Egyptian Fractions

The ancient Egyptians only used fractions of the form 1/n so any other fraction had to be represented as a sum of such unit fractions and, furthermore, all the unit fractions were different!
Why? Is this a better system than our present day one? In fact, it is for some tasks.

This page explores some of the history and methods with puzzles and and gives you a summary of computer searches for such representations. There's lots of investigations to do in this area of maths suitable for 8-10 year olds as well as older students and it is also designed as a resource for teachers and educators. The Calculators embedded in the page provide helpful resources for your number searches.

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Contents of this page
The Things To Do icon indicates a Things to do section of questions to start your own investigations. The calculator calculator icon indicates that there is a live interactive calculator in that section.

An Introduction to Egyptian Mathematics

Some of the oldest writing in the world is on a form of paper made from papyrus reeds that grew all along the Nile river in Egypt. map of upper Nile [The image is a link to David Joyce's site on the History of Maths at Clarke University.] The reeds were squashed and pressed into long sheets like a roll of wall-paper and left to dry in the sun. When dry, these scrolls could be rolled up and easily carried or stored.

Some of the papyrus scrolls date back to about 2000 BC, around the time of the construction of the larger Egyptian pyramids. Because there are deserts on either side of the Nile, papyrus scrolls have been well preserved in the dry conditions.

So what was on them do you think? How to preserve a body as a mummy? Maybe it was how to construct the extensive system of canals used for irrigation across Egypt or on storage of grain in their great storage granaries? Perhaps they tell how to build boats out of papyrus reeds which float very well because pictures of these boats have been found in many Egyptian tombs?
The surprising answer is that the oldest ones are about mathematics!

Henry Rhind and his Papyrus scroll

One of the papyrus scrolls, discovered in a tomb in Thebes, was bought by a 25 year old Scotsman, Henry Rhind at a market in Luxor, Egypt, in 1858. After his death at the age of 30, the scroll found its way to the British Museum in London in 1864 and remained there ever since, being referred to as the Rhind Mathematical Papyrus (or RMP for short).

Rhind papyrus So what did it say?

The hieroglyphs (picture-writing) on the papyrus were only deciphered in 1842 (and the Babylonian clay-tablet cuneiform writing was deciphered later that century).

It starts off by saying that the scribe "Ahmes" is writing it about 1600 BC but that he had copied it from "ancient writings" which, from his description of the Pharoah of that time dates it to 2000 BC or earlier. The picture is also a link so click on it to go to the St Andrews MacTutor biography of Ahmes.

Since early civilisations would need to predict the start of spring accurately in order to sow seeds, then a large part of such mathematical writing has applications in astronomy. Also, calculations were needed for surveying (geometry) and for building and for accounting and for sharing bread and beer (given as wages) amongst several workers.

On this page we will look at how the Egyptians of 4000 years ago worked with fractions.

Egyptian Fractions

The Egyptians of 3000 BC had an interesting way to represent fractions.
Although they had a notation for 1/2 and 1/3 and 1/4 and so on (these are called reciprocals or unit fractions since they are 1/n for some number n), their notation did not allow them to write 2/5 or 3/4 or 4/7 as we would today.

Instead, they were able to write any fraction as a sum of unit fractions where all the unit fractions were different.

For example,

3/4 = 1/2 + 1/4

6/7 = 1/2 + 1/3 + 1/42


A fraction written as a sum of distinct unit fractions is called an Egyptian Fraction.

Why use Egyptian fractions today?

Suppose you and 7 other friends go for a pizza. You all like the same kind but you don't want a whole one each. The 8 of you decide to buy 5 identical pizzas. How do you divide them up between you? Decimals don't help and that you each get 5/8 only tells you what the problem is (split 5 things into 8 parts) not the solution! It is easier with beer or sacks of grain. This was an everyday problem in ancient Egypt when barley loaves may have to be divided amongst workers.

A practical use of Egyptian Fractions

So suppose Fatima has 5 loaves of bread to share among the 8 workers who have helped dig her fields today and clear the irrigation channels. Pause for a minute and decide how YOU would solve this problem before reading on.....

HINT: What if there were only 4 loaves not 5 to be split amongst 8 people?
First Fatima sees that they all get at least half a loaf, so she uses 4 of the loaves to give all 8 of them half a loaf each. She has one whole loaf left.
Now it is easy to divide one loaf into 8, so they get an extra eighth of a loaf each and all the loaves are divided equally between the 5 workers. On the picture here they each receive one red part (1/2 a loaf) and one green part (1/8 of a loaf): 4 loaves split into halves and 1 split into eighths

and 5/8 = 1/2 + 1/8


The Egyptian solution has the added benefits of

Try the following using the Egyptian style of thinking:

Things to do /

  1. Suppose Fatima had 3 loaves to share between 4 people. How would she do it?
    3/4 = 1/2 + 1/4
  2. ...and what if it was 2 loaves amongst 5 people?
    2/5 = 1/2 + 1/15 is the simplest solution
  3. ...or 4 loaves between 5 people?
    4/5 = 1/2 + 1/4 + 1/20 or
    4/5 = 1/2 + 1/5 + 1/10
  4. What about 13 loaves to share among 12 people? We could give them one loaf each and divide the 13th into 13 parts for the final portion to give to everyone.
    Try representing 13/12 as 1/2 + 1/3 + 1/* . What does this mean - that is, how would you divide the loaves using this representation?
    Was this easier?
    12/13 = 1/2 + 1/3 + 1/12 + 1/156 12/13 = 1/2 + 1/3 + 1/13 + 1/78 12/13 = 1/2 + 1/4 + 1/6 + 1/156
It turns out that Egyptian fractions are not only a very pratical solution to some everyday problems today but are interesting in their own right. They had practical uses in the ancient Egyptian method of multiplying and dividing, and every fraction t/b can always be written as an Egyptian fraction, which we will show further down on this page. There are also many unsolved problems concerning them, which are still a puzzle to mathematicians today.

Calculator A Calculator to convert a Fraction to and from an Egyptian Fraction

An Egyptian Fraction for t/b is a sum of unit fractions, all different. Use the Shortest EF Calculator or the Fixed Length EFs Calculator to find all the Egyptian Fractions but this calculator is quicker if you just want one.
The method used in this calculator is the Greedy Algorithm which we will examine in more detail below. The disadvantage of the "greedy" method is that sometimes it will fail to fully convert a fraction if a denominator gets too large for the Calculator.

Fraction ↔ EF C A L C U L A T O R
Fraction Denominators
     
     
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R E S U L T S
calculator: Fraction ↔ EF Shortest EF Fixed Length EFs Factored EFs Harmonic Numbers

Different representations for the same fraction

We have already seen that 3/4 = 1/2 + 1/4
Can you write 3/4 as 1/2 + 1/5 + 1/* ?
What about 3/4 as 1/2 + 1/6 + 1/* ?
How many more can you find?

Here are some results that mathematicians have proved:

Now let's examine each of these in turn and I'll try to convince you that each is true for all fractions t/b less than one (so that T, the number on top, is smaller than B, the bottom number).

You can skip over these two sub-sections if you like.

Each fraction has an infinite number of Egyptian fraction forms

To see why the second fact is true, consider this:
1 = 1/2 + 1/3 + 1/6 (*)
So if
3/4 = 1/2 + 1/4
Let's use (*) to expand the final fraction 1/4:
So let's divide equation (*) by 4:
1/4 = 1/8 + 1/12 + 1/24
which we can then feed back into our Egyptian fraction for 3/4:
3/4 = 1/2 + 1/4
3/4 = 1/2 + 1/8 + 1/12 + 1/24
But now we can do the same thing for the final fraction here, dividing equation (*) by 24 this time. Since we are choosing the largest denominator to expand, it will be replaced by even larger ones so we won't repeat any denominators that we have used already:
1/24 = 1/48 + 1/72 + 1/144
and so
3/4 = 1/2 + 1/8 + 1/12 + 1/48 + 1/72 + 1/144
Now we can repeat the process by again expanding the last term: 1/144 and so on for ever!
Each time we get a different set of unit fractions which add to 3/4!
This shows conclusively once we have found one way of writing t/b as a sum of unit fractions, then we can derive as many other representations as we wish! If T=1 already (so we have 1/B) then using (*) we can always start off the process by dividing (*) by B to get an initial 3 unit fractions that sum to 1/B.

Every ordinary fraction has an Egyptian Fraction form

We now show there is always at least one set of distinct unit fractions which sum to any given fraction t/b≤1 by actually showing how to find such a sum.

Fibonacci's Greedy Algorithm for finding Egyptian Fractions

This method and a proof are given by Fibonacci in his book Liber Abaci produced in 1202, the book in which he mentions the rabbit problem involving the Fibonacci Numbers. It is the method used in the Fraction ↔ EF CALCULATOR above.

Remember that

The method is to find the biggest unit fraction we can and take it from t/b and hence its other name - the greedy algorithm.
With what is left, we repeat the process. We will show that this series of unit fractions always decreases, never repeats a fraction and eventually will stop. Such processes are now called algorithms and this is an example of a greedy algorithm since we (greedily) take the largest unit fraction we can and then repeat on the remainder.

Let's look at an example before we present the proof: 521/1050.
521/1050 is less than one-half (since 521 is less than a half of 1050) but it is bigger than one-third. So the largest unit fraction we can take away from 521/1050 is 1/3:

521/1050 = 1/3 + R
What is the remainder?
521/1050 - 1/3 = 171/1050 = 57/350
So we repeat the process on 57/350:
This time the largest unit fraction less than 57/350 is 1/7 and the remainder is 1/50.
How do we know it is 7? Divide the bottom (larger) number, 350, by the top one, 57, and we get 6.14... . So we need a number larger than 6 (since we have 6 + 0.14) and the next one above 6 is 7.)
So
521/1050 = 1/3 + 57/350
= 1/3 + 1/7 + 7/350
= 1/3 + 1/7 + 1/50
The sequence of remainders is important in the proof that we do not have to keep on doing this for ever for some fractions t/b:
521/1050, 171/1050, 7/1050
The numerator of the remainder is getting smaller each time. If it keeps decreasing then it must eventually reach 1 and the process stops. In this example, we find the numerator becomes a factor of the denominator and so the final remainder is a unit fraction and we can stop.

Practice with these examples and then we'll have a look at finding short Egyptian fractions.

Things to do /

The Greedy method is used in the Fraction ↔ EF CALCULATOR above
  1. What does the greedy method give for 5/21?
    What if you started with 1/6 (what is the remainder)?
    5/12 = 1/5 + 1/27 + 1/945 by the Greedy method but
    5/21 = 1/6 + 1/14
  2. Can you improve on the greedy method's solution for 9/20 (that is, use fewer unit fractions)? [Hint: Express 9 as a sum of two numbers which are factors of 20.]
    9/20 = 1/3 + 1/9 + 1/180
    9/20 = 1/4 + 1/5
  3. The numbers in the denominators can get quite large using the greedy method: What does the greedy method give for 5/91?
    Can you find a two term Egyptian fraction for 5/91?
    [Hint: Since 91 = 7x13, try unit fractions which are multiples of 7.]
    5/91 = 1/19 + 1/433 + 1/249553 + 1/93414800161 + 1/too large
    5/91 = 1/28 + 1/52

A Proof

This section is optional: click on the button see the proof.
Now let's see how we can show this is true for all fractions t/b. We want
t/b= 1/u1 + 1/u2 + ... + 1/un
where u1 < u2 < ... < un
Also, we are choosing the largest u1 at each stage, hence the name of "the greedy algorithm".
What does this mean?
It means that
1/u1 < t/b
but that 1/u1 is the largest such fraction. For instance, we found that 1/3 was the largest unit fraction less than 521/1050. This means that 1/2 would be bigger than 521/1050.
In general, if 1/u1 is the largest unit fraction less than t/b then
1/u1-1 > t/b

To find the largest denominator, evaluate b/t and use that value if it is an integer, or else round it up if not.

Since t>1, neither 1/u1 nor 1/u1-1 equal t/b.

What is the remainder?

It is

t/b1/u1 = (t*u1 – b)/(b*u1)
Also, since
1/(u1-1) > t/b
then multiplying both sides by b we have
b/(u1-1) > t
or, multiplying both sides by (u1-1) and expanding the brackets, then adding t and subtracting b from both sides we have:
b > t (u1 - 1)
b > t u1 - t
t > t u1 - b
Now t*u1 – b was the numerator of the remainder and we have just shown that it is smaller than the original numerator t. If the remainder, in its lowest terms, has a 1 on the top, we are finished. Otherwise, we can repeat the process on the remainder, which has a smaller denominator and so the remainder when we take off its largest unit fraction gets smaller still. Since t is a whole (positive) number, this process must inevitably terminate with a numerator of 1 at some stage.

That completes the proof that


The next section explores the shortest Egyptian fractions for any given fraction.

Shortest Egyptian Fractions

The greedy method and the shortest Egyptian fraction

However, the Egyptian fraction produced by the greedy method may not be the shortest such fraction. Here is an example:
by the greedy method, 4/17 reduces to
4/17 = 1/5 + 1/29 + 1/1233 + 1/3039345
whereas we can also check that
4/17 = 1/5 + 1/30 + 1/510

Here is the complete list of all the shortest representations of t/b for b up to 11. We use a list notation here to make the unit fractions more readable. For instance, above we saw that:

4/5 = 1/2 + 1/4 + 1/20
which we will write as:
4/5 = [2, 4, 20]
2/3 = [2, 6]
2/5 = [3, 15]
2/7 = [4, 28]
2/9 = [5, 45] = [6, 18]
2/11 = [6, 66]
3/4 = [2, 4]
3/5 = [2, 10]
3/7 = [3, 11, 231] = [3, 12, 84] = [3, 14, 42] = [3, 15, 35] = [4, 6, 84] = [4, 7, 28]
3/8 = [3, 24] = [4, 8]
3/10 = [4, 20] = [5, 10]
3/11 = [4, 44]
4/5 = [2, 4, 20] = [2, 5, 10]
4/7 = [2, 14]
4/9 = [3, 9]
4/11 = [3, 33]
5/6 = [2, 3]
5/7 = [2, 5, 70] = [2, 6, 21] = [2, 7, 14]
5/8 = [2, 8]
5/9 = [2, 18]
5/11 = [3, 9, 99] = [3, 11, 33] = [4, 5, 220]
6/7 = [2, 3, 42]
6/11 = [2, 22]
7/8 = [2, 3, 24] = [2, 4, 8]
7/9 = [2, 4, 36] = [2, 6, 9]
7/10 = [2, 5]
7/11 = [2, 8, 88] = [2, 11, 22]
8/9 = [2, 3, 18]
8/11 = [2, 5, 37, 4070] = [2, 5, 38, 1045] = [2, 5, 40, 440] = [2, 5, 44, 220] = [2, 5, 45, 198] = [2, 5, 55, 110] = [2, 5, 70, 77] = [2, 6, 17, 561] = [2, 6, 18, 198] = [2, 6, 21, 77] = [2, 6, 22, 66] = [2, 7, 12, 924] = [2, 7, 14, 77] = [2, 8, 10, 440] = [2, 8, 11, 88] = [3, 4, 7, 924]
9/10 = [2, 3, 15]
9/11 = [2, 4, 15, 660] = [2, 4, 16, 176] = [2, 4, 20, 55] = [2, 4, 22, 44] = [2, 5, 10, 55]
10/11 = [2, 3, 14, 231] = [2, 3, 15, 110] = [2, 3, 22, 33]
8/11 has an unusually large number of different (shortest) representations!

Calculator A Calculator for Shortest Egyptian Fractions

The calculator below will find all the Egyptian fractions of shortest length for an ordinary fraction.
Shortest EF C A L C U L A T O R
  the shortest Egyptian fractions for
  
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R E S U L T S

calculator: Fraction ↔ EF Shortest EF Fixed Length EFs Factored EFs Harmonic Numbers

The number of Shortest Length Egyptian Fractions

Here is a table of the lengths of the shortest Egyptian Fractions for all fractions t/b (Top over Bottom) where the denominator B takes all values up to 30:

Shortest Egyptian Fractions lengths for fraction T/B

KEY:
means the fraction t/b is not in its lowest form e.g. 9/12 so find its lowest form P/Q (9/12=3/4) and then look up that fraction
. means the fraction t/b is bigger than 1. Try B/T instead!
number is the minimum number of unit fractions that are needed to sum to t/b.
Find T (top or numerator) down the side and B (bottom or denominator) across the top.
\B:                1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3
T\   3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
 2|  2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 -
 3|  . 2 2 - 3 2 - 2 2 - 3 2 - 2 2 - 3 2 - 2 2 - 2 2 - 2 2 -
 4|  . . 3 - 2 - 2 - 2 - 3 - 2 - 3 - 2 - 2 - 2 - 3 - 2 - 3 -
 5|  . . . 2 3 2 2 - 3 2 3 2 - 2 3 2 2 - 2 3 3 2 - 2 2 2 2 -
 6|  . . . . 3 - - - 2 - 3 - - - 2 - 3 - - - 2 - 2 - - - 2 -
 7|  . . . . . 3 3 2 3 2 2 - 3 3 3 2 3 2 - 3 3 2 3 2 2 - 3 2
 8|  . . . . . . 3 - 4 - 3 - 2 - 4 - 3 - 2 - 2 - 3 - 3 - 3 -
 9|  . . . . . . . 3 4 - 3 2 - 2 2 - 4 2 - 3 3 - 3 2 - 2 3 -
10|  . . . . . . . . 4 - 3 - - - 3 - 2 - 2 - 3 - - - 2 - 2 -
11|  . . . . . . . . . 3 3 3 3 3 3 2 3 2 2 - 3 2 3 3 3 2 3 2
12|  . . . . . . . . . . 4 - - - 3 - 3 - - - 2 - 4 - - - 4 -
13|  . . . . . . . . . . . 4 3 3 3 3 3 3 3 2 3 2 2 - 3 3 3 2
14|  . . . . . . . . . . . . 3 - 4 - 4 - - - 3 - 3 - 2 - 4 -
15|  . . . . . . . . . . . . . 4 4 - 4 - - 3 4 - - 2 - 2 2 -
16|  . . . . . . . . . . . . . . 5 - 3 - 3 - 4 - 3 - 3 - 3 -
17|  . . . . . . . . . . . . . . . 3 4 3 3 3 4 3 3 3 3 3 3 2
18|  . . . . . . . . . . . . . . . . 4 - - - 4 - 3 - - - 4 -
19|  . . . . . . . . . . . . . . . . . 3 3 3 4 3 3 4 3 3 4 3
20|  . . . . . . . . . . . . . . . . . . 4 - 4 - - - 4 - 4 -
21|  . . . . . . . . . . . . . . . . . . . 4 5 - 3 4 - - 4 -
22|  . . . . . . . . . . . . . . . . . . . . 5 - 4 - 4 - 3 -
23|  . . . . . . . . . . . . . . . . . . . . . 3 4 4 3 3 4 3
24|  . . . . . . . . . . . . . . . . . . . . . . 4 - - - 4 -
25|  . . . . . . . . . . . . . . . . . . . . . . . 4 4 3 4 -
26|  . . . . . . . . . . . . . . . . . . . . . . . . 4 - 4 -
27|  . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 -
28|  . . . . . . . . . . . . . . . . . . . . . . . . . . 5 -
29|  . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Are there fractions whose shortest EF length is 3 (4, 5, ..) ?

From the table above, we see the "smallest" fraction that needs three terms is T=4 B=5 i.e. 4/5
In fact there are two ways to write 4/5 as a sum of three unit fractions:
4/5 = 1/2 + 1/4 + 1/20 and 4/5 = 1/2 + 1/5 + 1/10

There are many other fractions whose shortest EF has 3 unit fractions. Those with a denominator 10 or less are:

4/5 3/7 5/7 6/7 7/8 7/9 8/9 9/10

Is there a fraction that needs 4 unit fractions?
Yes! 8/11 canot be written as a sum of less than 4 unit fractions, for instance

8/11 = 1/2 + 1/6 + 1/22 + 1/66
and there are 15 other EFs of length 4 for this fraction.
Other fractions with a denominator 20 or less that need at least 4 unit fractions are:
8/11 9/11 10/11 12/13 13/14 15/16 8/17 14/17 15/17 9/19 14/19 15/19 17/19 18/19

This leads us naturally to ask:
Is there a fraction that needs 5 unit fractions?
Yes! The smallest numerator and denominator are for the fraction 16/17

16/17 = 1/2 + 1/3 + 1/17 + 1/34 + 1/51
and there are 38 other EFs of length 5 for this fraction.
Other fractions with a denominator 40 or less that need 5 unit fractions are:
16/17 21/23 22/23 27/29 28/29 30/31 32/34 33/34 36/37 37/38 38/39

Continuing:
The smallest fraction needing 6 unit fractions is 77/79

77/79 = 1/2 + 1/3 + 1/8 + 1/79 + 1/474 + 1/632
and there are 159 other EFs of length 6 for this fraction.
Other fractions with a denominator up to 130 that need 6 unit fractions are:
77/79 101/107 102/103 104/107 106/107 108/109 112/113 115/118 117/118 119/127 123/127

The smallest fraction needing 7 unit fractions is 732/733

732/733 = 1/2 + 1/3 + 1/7 + 1/45 + 1/7330 + 1/20524 + 1/26388
seems to be the one with the smallest numbers (the maximum denominator is the least among all the solutions) according to David Epstein in this MathForum post of March 2000.
He also states that he found 2771 separate 7-term EFs for this fraction.

The smallest fraction needing 8 unit fractions is 27538/27539.
Mr. Huang Zhibin () of China in April 2014 has verified that this fraction needs 8 unit fractions and gives this example:

27538/27539 = 1/2+1/3+1/7+1/43+1/1933+1/14893663+1/1927145066572824+1/212829231672162931784
Beyond 8 unit fractions is unknown territory!

Finding patterns for shortest Egyptian Fractions

There seem to be lots of patterns to spot in the table above.
The top row, for instance, seems to have the pattern that 2/B can be written as a sum of just 2 unit fractions (providing that B is odd since otherwise, 2/B would not be in its "lowest form"). The odd numbers are those of the form 2i+1 as i goes from 1 upwards. Let's list some of these in full:
i2/(2i+1)
1 2/3 = 1/2 + 1/6
2 2/5 = 1/3 + 1/15
3 2/7 = 1/4 + 1/28
4 2/9 = 1/5 + 1/45
2/9 = 1/6 + 1/18
5 2/11 = 1/6 + 1/66
6 2/13 = 1/7 + 1/91
7 2/15 = 1/8 + 1/120
2/15 = 1/9 + 1/45
2/15 = 1/10 + 1/30
2/15 = 1/12 + 1/20
Let's concentrate on the first sum on each line since some of the fractions above have more than one form as a sum of two unit fractions.

It looks as if

2/2i+1 = 1/i+1 + 1/?
Can you spot how we can use (2i+1) and i to find the missing number?
Here is the table again with the (2i+1) and i+1 parts in red and the ? number is in green :
i2/2i+1 = 1/i+1 + 1/?
1 2/3 = 1/2 + 1/6
2 2/5 = 1/3 + 1/15
3 2/7 = 1/4 + 1/28
4 2/9 = 1/5 + 1/45 = 1/6 + 1/18
5 2/11 = 1/6 + 1/66
6 2/13 = 1/7 + 1/91
7 2/15 = 1/8 + 1/120 = 1/9 + 1/45 = 1/10 + 1/30 = 1/12 + 1/20
8 2/17 = 1/9 + 1/153
9 2/19 = 1/10 + 1/190
Yes! Just multiply the red numbers i+1 and 2i+1 to get the green ones!

So it looks like we may have the pattern:
2/2i+1 = 1/i+1 + 1/(i+1)(2i+1)
We can check it by simplifying the fraction on the right and seeing if it reduces to the one on the left:
1/i+1 + 1/(i+1)(2i+1) = (2i+1 + 1)/(i+1)(2i+1) = 2i+2/(i+1)(2i+1) = 2(i+1)/(i+1)(2i+1) = 2/2i+1
So algebra has shown us that the formula is always true.

How many Egyptian Fractions of shortest length are there for T/B?

Here is a table like the one above, but this time each entry is a count of all the ways we can write t/b as a sum of the minimum number of unit fractions:
For instance, we have seen that 4/5 can be written with a minimum of 2 unit fractions, so 2 appears in the first table under t/b=4/5.
But we saw that 4/5 has two ways in which it can be so written, so in the following table we have entry 2 under t/b=4/5.
2/15 needs at least 2 unit fractions in its Egyptian form: here are all the variations:
2/15 = 1/8 + 1/120
= 1/9 + 1/45
= 1/10 + 1/30
= 1/12 + 1/20
so it has four representations. In the table below, under t/b=2/15 we have the entry 4:

NUMBER of Shortest Egyptian Fractions:
\B  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
T\                                                                                       
 2  .  1  -  1  -  1  -  2  -  1  -  1  -  4  -  1  -  1  -  4  -  1  -  2  -  3  -  1  -
 3  .  .  1  1  -  6  2  -  2  1  -  8  3  -  2  1  -  8  4  -  2  1  -  1  3  -  3  1  -
 4  .  .  .  2  -  1  -  1  -  1  -  4  -  3  -  4  -  1  -  2  -  1  - 10  -  2  -  7  -
 5  .  .  .  .  1  3  1  1  -  3  2  5  1  -  1  5  2  1  -  1 15  8  3  -  1  1  2  1  -
 6  .  .  .  .  .  1  -  -  -  1  -  1  -  -  -  1  -  2  -  -  -  1  -  1  -  -  -  1  -
 7  .  .  .  .  .  .  2  2  1  2  2  1  -  7  5  3  1  5  2  -  5  3  2 10  1  1  -  2  2
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10  .  .  .  .  .  .  .  .  .  3  -  1  -  -  -  3  -  1  -  1  -  1  -  -  -  1  -  1  -
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12  .  .  .  .  .  .  .  .  .  .  .  3  -  -  -  1  -  1  -  -  -  1  - 47  -  -  - 29  -
13  .  .  .  .  .  .  .  .  .  .  .  .  6  2  1  1  2  1  5  4  1  2  1  1  -  2  5  1  1
14  .  .  .  .  .  .  .  .  .  .  .  .  .  1  -  2  -  5  -  -  -  1  -  4  -  1  - 10  -
15  .  .  .  .  .  .  .  .  .  .  .  .  .  .  5  4  -  3  -  -  1 13  -  -  1  -  1  1  -
16  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 39  -  1  -  1  -  7  -  1  -  4  -  2  -
17  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  3  2  1  1  2  4  1  1  2  4  2  1
18  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  -  -  -  5  -  1  -  -  - 14  -
19  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  1  1  1  2  1  8  2  2  6  6
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27  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  5 20  -
28  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 16  -
29  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  7

Fixed Length Egyptian Fractions

The shortest Egyptian fractions do not always give the smallest numbers.
For example, the smallest number of unit fractions for 8/11 is 4; there are 16 of them and the one with the smallest numbers (i.e. the one whose largest denominator is the smallest) is 8/11 = 1/2 + 1/6 + 1/22 + 1/66
However, if we look for a larger collection, of 5 unit fractions, we find smaller numbers still:
8/11 = 1/2 + 1/11 + 1/12 + 1/33 + 1/44 and
8/11 = 1/3 + 1/4 + 1/11 + 1/33 + 1/44

Here we investigate Egyptian Fractions with more than the smallest number of reciprocals.
We have already seen that every fraction t/b has an Egyptian Fraction and, what is more, an infinite number of longer and longer Egyptian fraction forms.
So let's see what we can find about the number of Egyptian fractions for t/b of a given length L.

Calculator A Fixed Length Egyptian Fraction Calculator

Fixed Length EF C A L C U L A T O R
Options: leave blank if not needed


Egyptian fractions of length for
--
all denominators ≤          
find no more than
solutions
R E S U L T S

calculator: Fraction ↔ EF Shortest EF Fixed Length EFs Factored EFs Harmonic Numbers

Egyptian Fractions for 1

Earlier we used an Egyptian Fraction for 1 in a proof that every EF can be expanded into an infinite number of alternative sums for the same fraction. We used:
1 = 1/2 + 1/3 + 1/6
This is both the only way to represent 1 as a sum of unit fractions, it is the shortest and its maximum denominator (6) is the least possible. Before you read on, you might like to try and find 4 different unit fractions that sum to 1.
Hint:
6 = 3 + 2 + 1 and since it is a sum of different divisors of 6, we can divide by 6 to find our 3-term EF for 1 shown above.
Can you find another number which is also a sum of some of its divisors?
What about 12 = 6 + 4 + 2?
Dividing by 12 gives us 1 = 1/2 + 1/3 + 1/6 which we already know.

Here is a fixed length EF for 1 that uses 4 unit fractions:
12 = 6 + 3 + 2 + 1 Dividing by 12 gives 1 = 1/2 + 1/4 + 1/6 + 1/12
Other solutions are found from
18 = 9 + 6 + 2 + 1> ⇔ 1 = 1/2 + 1/3 + 1/9 + 1/18
20 = 10 + 5 + 4 + 1 ⇔ 1 = 1/2 + 1/4 + 1/5 + 1/20
24 = 12 + 8 + 3 + 1 ⇔ 1 = 1/2 + 1/3 + 1/8 + 1/24
30 = 15 + 10 + 3 + 2 ⇔ 1 = 1/2 + 1/3 + 1/10 + 1/15
42 = 24 + 14 + 6 + 1 ⇔ 1 = 1/2 + 1/3 + 1/7 + 1/42
so there are 6 EFs for 1 of length 4.

How about 5 different unit fractions that sum to 1?
There are more than 10 but less than 100.

Use the Calculator above to see all 72 solutions
The one with smallest denominators is 1 = 1/2 + 1/4 + 1/10 + 1/12 + 1/15
We found: The number of ways to write 1 as a sum of 3, 4, 5, 6, 7 and 8 different unit fractions is
1, 6, 72, 2320, 245765, 151182379 which is A006585
We do not know any more terms in this series as the only approach so far has been to perform a computer search and the number of solutions is getting very large very quickly!
A table of the solutions for smaller lengths is given in A073546 but the Calculator above will find these for you.

Let's now look at some special forms of EFs for 1.

EFs for 1 which begin 1/n

It seems there is always an EF for 1 which begins with 1/n for any n we like!
One such expression for 1 as a sum of distinct unit fractions of which the largest is 1/n is always guaranteed to exist for any and every n>1 as was shown by Botts in 1967: Starting from 1=1/2+1/3+1/6, he uses the Expansion Equation:
1=1+1
nn + 1n ( n + 1)
to replace the smallest 1/n by the right-hand side of this equation. If any duplicates arise, he uses it again on one of the duplicates.
He proves that by repeating this process we can eventually remove all duplicates with any value 1/n as the largest unit fraction.
However the resulting EFs become quite long quite quickly:
DenominatorsLength
2, 3, 63
3, 4, 6, 7, 12, 426
4, 5, 6, 7, 12, 13, 20, 42, 1569
5, 6, 7, 8, 12, 13, 20, 21, 30, 42, 43, 56, 156, 420, 180615
6, 7, 8, 9, 12, 13, 20, 21, 30, 31, 42, 43, 44, 56, 57, 72, 156, 420, 930, 1806, 1807, 1892, 3192, 326344224
7, 8, 9, 10, 12, 13, 20, 21, 30, 31, 42, 43, 44, 45, 56, 57, 58, 72, 73, 90, 156, 420, 930, 1806, 1807, 1808, 1892, 1893, 1980, 3192, 3193, 3306, 5256, 3263442, 3263443, 3267056, 3581556, 10192056, 1065005695080639
Two questions now arise
Is there a method to produce shorter EFs for 1 beginning with 1/n?
Is there a method to produce smaller denominators than the above?
Here is a table of some of the shorter EFs for 1 beginning with 1/n:
n Denominators
22, 3, 6
33, 4, 6, 10, 12, 15
44, 5, 6, 9, 10, 15, 18, 20
55, 6, 8, 9, 10, 12, 15, 18, 20, 24
66, 7, 8, 9, 10, 12, 14, 15, 18, 24, 28
77, 8, 9, 10, 11, 12, 14, 15, 18, 22, 24, 28, 33
88, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 28, 30, 33, 35, 40
These are not the shortest: 3, 4, 5, 6, 20 is shorter starting with 3 but maybe they have the smallest (maximum) denominators?
Can you find any shorter or with smaller numbers?

EFs for 1 with even denominators

Look at the following table of denominators of EFs for 1 which are all even, each having one more unit fraction than the previous one:
2, 4, 6, 12
2, 4, 6, 14, 84
2, 4, 6, 14, 86, 3612
2, 4, 6, 14, 86, 3614, 6526884
2, 4, 6, 14, 86, 3614, 6526886, 21300113901612
2, 4, 6, 14, 86, 3614, 6526886, 21300113901614, 226847426110843688722000884
There seems to be a pattern here:
The last denominator 1/(2D) can be replaced by two new ones, the smallest of which is 1/(2D+2) and both the old and the two new denominators are even.
A little algebra shows why:
1 = 1 + 1   ..... Let's call this the Expand Even Rule
2D 2D + 2 2D(D + 1)
The expansion preserves the evenness of the denominators.
We can continue this expansion and so show that
There is always an expansion of 1 as an EF with distinct even denominators of any length greater than 3.

Things to do /

Above we expanded the final even denominator. This section helps you to explote what happens if we expanded the others using the Expand Even Rule?
  1. If we start from 2,2 as the denominators of our initial EF for 1, what does the Rule give if we expand one of the 2's?
    2, 4, 4
  2. What about expanding 4?
    6,12

    If we start with the EF denominators 2, 4, 4 and expanded the final 4, which EF for 1 do we get?
    2, 4, 6, 12
  3. Starting from 2, 4, 6, 12 expand the 6 to get a new EF for 1.
    2, 4, 8, 12, 24
  4. In your previous answer, expand the 8 or one of the others. Which EFs do you get?
    2, 4, 8, 12, 24 -> 4, 4, 4, 8, 12, 24
    2, 4, 8, 12, 24 -> 2, 6, 12, 12, 24
    2, 4, 8, 12, 24 -> 2, 4, 10, 40, 12, 24
    2, 4, 8, 12, 24 -> 2, 4, 8, 14, 84, 24
    2, 4, 8, 12, 24 -> 2, 4, 8, 12, 26, 312

EFs for 1 with odd denominators

When it comes to all denominators being odd, the problem is quite different! There are none of lengths 2, 3, 4, 5, 6, 7 or 8!
The first has 9 unit fractions and there are 5 of them with denominators as follows:
3, 5, 7, 9, 11, 15, 35, 45, 231
3, 5, 7, 9, 11, 15, 21, 231, 315
3, 5, 7, 9, 11, 15, 33, 45, 385
3, 5, 7, 9, 11, 15, 21, 165, 693
3, 5, 7, 9, 11, 15, 21, 135, 10395
A computer search reveals that there are none of length 10.
There do not seem to be any EFs for 1 with an even number of odd denominators.
Some thought about evenness and oddness (parity) reveals why there never will be: Can you find a reason why the length must be odd?
The proof uses a parity argument that is, it uses just the oddness and evenness of numbers.
Any odd number can be written as 2n + 1 and every even number is of the form 2n
a SUM of two odd numbers is even:
2n + 1 + 2m + 1 = 2n + 2m + 2 = 2(n + m + 1)
a sum of an even number of odds is always even
a PRODUCT of two odd numbers is odd:
(2n + 1)(2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1
a product of any number of odd numbers is always odd.
Therefore an odd number can never have an even factor. Only even numbers have even factors.
Consider just two unit fractions with odd denominator that sum to 1 and add them to get a single fraction as follows
1 = 1 + 1 = odd1 + odd2
odd1odd2odd1×odd2
The numerator is a sum of two odds and so is even.
The denominator is a product of 2 odds and so is odd.
So the numerator and denominator cannot be identical and the fraction cannot be 1.

The same thing happens if we have a sum of 4 unit fractions with odd denominators:
The numerator is a sum of an even number of odds and so must be even
the collected denominator is a product of odd numbers and is therefore odd.
So the fraction can never be reduced to 1 when there are an even number of unit fractions summing to 1.

Thus an EF for 1 which has only odd denominators can never be of even length.

There are EFs for 1 with odd denominators and of odd length from 11 as shown here:

length 11
3, 5, 7, 9, 15, 21, 27, 35, 63, 105, 135
length 13
3, 5, 7, 9, 15, 21, 35, 45, 63, 99, 105, 143, 195
length 15
3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 135, 189, 255, 323, 399
length 17
3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 135, 255, 323, 399, 483, 575, 675
length 19
3, 5, 7, 11, 15, 21, 35, 45, 63, 77, 91, 143, 209, 273, 285, 315, 399, 441, 931
length 21
3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 165, 231, 255, 323, 399, 483, 575, 675, 783, 899, 1023
length 23
3, 5, 7, 11, 15, 21, 35, 45, 63, 77, 105, 165, 231, 255, 323, 399, 483, 575, 675, 783, 899, 1023, 1155
...
so it seems likely these exist for every odd length from 9 upwards....
...and we can prove this if, as with proving that there are an infinite number of EFs derivable from any EF in the section above, we can find an expansion formula for 1/odd that uses 3 odd denominators.
Try to find one yourself or else
For instance a bit of algebraic manipulation will show that:
1 =  1 + 1 + 1
2n + 12n + 3(n + 1) (2 n + 3) – 1(n + 2) (2n + 1) (2 n + 3)
If n itself is odd and only then will each of the 3 denominators on the right be odd too.

So now we know:

There will always be odd length EF for 1 consisting of only odd denominators for all odd lengths from 9 onwards.
What we still don't know is how many there are of odd length n for n from 11 onwards.
Nor indeed do we know the ones with the smallest numbers, that is, the largest denominator in the EFs for 1 of a given (odd) length which is the smallest possible.

EFs for 1 with the smallest denominators

Here is a list of the EF's for 1 of various lengths with smallest denominators where now we allow any number as denominator but still maintaining the property that all the denominators are different:
Length Denominators with sum of 1
32 3 6
42 4 6 12
52 4 10 12 15
63 4 6 10 12 15
73 4 9 10 12 15 18
83 5 9 10 12 15 18 20
4 5 6 9 10 15 18 20
9 4 5 8 9 10 15 18 20 24
4 6 8 9 10 12 15 18 24
10 5 6 8 9 10 12 15 18 20 24
11 5 6 8 9 10 15 18 20 21 24 28
5 7 8 9 10 14 15 18 20 24 28
6 7 8 9 10 12 14 15 18 24 28
12 4 8 9 10 12 15 18 20 21 24 28 30
6 7 8 9 10 14 15 18 20 24 28 30
13 4 8 9 11 12 18 20 21 22 24 28 30 33
4 8 10 11 12 15 20 21 22 24 28 30 33
4 9 10 11 12 15 18 20 21 22 28 30 33
5 8 9 10 11 12 18 21 22 24 28 30 33
5 8 9 10 11 15 18 20 21 22 24 28 33
6 7 8 9 11 14 18 20 22 24 28 30 33
6 7 8 10 11 14 15 20 22 24 28 30 33
6 7 9 10 11 14 15 18 20 22 28 30 33
6 8 9 10 11 12 15 18 20 22 24 30 33
6 8 9 10 11 12 15 18 21 22 24 28 33
7 8 9 10 11 12 14 15 18 22 24 28 33
146 8 9 10 11 15 18 20 21 22 24 28 30 33
7 8 9 10 11 14 15 18 20 22 24 28 30 33
156 8 9 11 14 15 18 20 21 22 24 28 30 33 35
166 8 11 12 14 15 18 20 21 22 24 28 30 33 35 36
176 10 11 12 14 15 18 20 21 22 24 28 30 33 35 36 40
187 10 11 12 14 15 18 20 21 22 24 28 30 33 35 36 40 42
So of all the EFs for 1 with 3 fractions, the smallest has all denominators no bigger than 6.
Of those EFs for 1 with 4 fractions, the smallest has no denominator bigger than 12. and for 5 fractions, the smallest has no denominator bigger than 15.
The series of these smallest maximum denominators (the minimax solution) in the EFs for 1 of various lengths is given by:
6, 12, 15, 15, 18, 20, 24, 24, 28, 30, 33, 33, 35, 36, 40, 42, ... A030659.

Things to do /

  1. Suppose we find list the EFs of 1 arranged in order of the sums of their denominators.
    This list starts with 11 since 1 = 1/2 + 1/3 + 1/6 and 2+3+6=11
    The next numbers are 24 and then 30, 31 and 32. How does the list continue?
    Check your answers with A051907 Note that some denominator sums are repeated since
    1 = 1/2 + 1/4 + 1/9 + 1/12 + 1/18
    1 = 1/2 + 1/5 + 1/6 + 1/12 + 1/20
    and both have a denominator sum of 45.
  2. Suppose we now allowed unit fractions which sum to 1 to have repeated unit fractions e.g.
    1 = 1/2 + 1/4 + 1/4 = 1/3 + 1/3 + 1/3
    There is now a total of 14 ways to write 1 as a sum of 4 unit fractions including those solutions we found above. What are they?
    Check your answers with A002966
  3. Fibonacci's Greedy algorithm to find Egyptian fractions with a sum of 1 is as follows:
    Choose the largest unit fraction we can, write it down and subtract it
    Repeat this on the remainder until we find the remainder is itself a unit fraction not equal to one already written down.
    At this point we could stop or else continue splitting the unit fraction into smaller fractions.
    To use this method to find a set of unit fractions that sum to 1:
    So we would start with 1/2 as the largest unit fraction less than 1:
    1 = 1/2 + ( 1/2 remaining)
    so we repeat the process on the remainder: the largest fraction less than 1/2 is 1/3:
    1 = 1/2 + 1/3 + ( 1/6 remaining).
    We could stop now or else continue with 1/7 as the largest unit fraction less than 1/6 ...
    1 = 1/2 + 1/3 + 1/7 + ...
    Find a few more terms, choosing the largest unit fraction at each point rather than stopping.
    The infinite sequence of denominators is called Sylvester's Sequence.
    Check your answers at A000058 in Sloane's Online Encyclopedia of Integer Sequences.
  4. Investigate shortest Egyptian fractions for 3/n:
    1. Find a fraction of the form 3/n that is not a sum of two unit fractions.
    2. Is it always possible to write 3/n as a sum of three unit fractions ?
      Give a formula for the different cases to verify your answer.
  5. Find a value for n where 4/n cannot be expressed as a sum of two unit fractions.

The Inheritance Puzzle

This is an old puzzle mentioned in many puzzle books in various guises:
A man who had 12 horses and 3 children wrote his will to leave 1/2 of his horses to Pat, 1/3 to Chris and 1/12 to Sam.
However, just after he died one of his horses died too.
How were the children to divide the 11 remaining horses so as to fulfil the terms of the will?
The answer is that a friend calls by and offers to add his own horse to the 11 others.
Now they can split the horses with a total of 11 horses exactly as specified in the will but leaving one horse over - the horse the friend brought. The friend could leave taking his horse with him and the children too go home happy.
What happened here?
The answer lies in the peculiar conditions of the will since 1/2 + 1/3 + 1/12 = 11/12 and not 12/12!
It works because the denominators 2, 3 and 12 are all factors of 12.

Here is another variation on the same inheritance puzzle:

A cactus collector had 11 rare cactii and in her will she left 1/2 of the collection to one other collector, 1/4 to another collector and 1/6 to a third. When she died how could the plants be divided as specified without splitting any?
The same solution applies, with the loan of one extra plant which is then returned on successful division of the 12 and the collectors receive 12/2 = 6, 12/4 = 3 and 12/6 = 2 and of course 6 + 3 + 2 = 11.

This kind of problem is designed from solutions to EFs for 1 where the denominators are all factors of their sum plus 1.
For instance in the cactii puzzle 11 = 6 + 3 + 2 and 6, 3, 2 are all factors of 11 + 1 = 12.
If we can divide these factors into 12 and also include 1/12 then we have an EF for 1 as follows:
1 = 12/12 = 12/6 + 12/3 + 12/2 + 1/12 = 1/2 + 1/4 + 1/6 + 1/12 but note that not all EFs for 1 have this form:
24, 8, 3, 2 are all divisors of 24 and dividing each into 24 gives 1/24 + 1/8 + 1/3 + 1/2 = 1 but 8 + 3 + 2 is only 13.

Numbers which are the sum of a subset of their divisors are called pseudo perfect numbers, named by Sierpinski in 1965.
For example, the smaller ones are:

nDivisors
with sum n
Divisors/n
61, 2, 31/2 + 1/3 + 1/2 = 1
122, 4, 61, 2, 3, 61/12 + 1/6 + 1/4 + 1/2 = 1
183, 6, 91, 2, 6, 91/18 + 1/9 + 1/3 + 1/2 = 1
201, 4, 5, 101/20 + 1/5 + 1/4 _ 1/2 = 1
The list of pseudo perfect (or semiperfect) numbers starts: 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66,... A005835

Things to do /

  1. There are a total of 7 possible puzzles like the horses and the cactus collection above if there are 3 inheritors.
    What are they?
    number
    to share
    ProportionsShares with
    1 loaned
    71/2, 1/4, 1/84, 2, 1
    111/2, 1/3, 1/126, 4, 1
    111/2, 1/4, 1/66, 3, 2
    171/2, 1/3, 1/99, 6, 2
    191/2, 1/4, 1/5 10, 5, 4
    231/2, 1/3, 1/8 12 ,8, 3
    411/2, 1/3, 1/721, 14, 6
  2. Suppose there are 23 horses and any number of inheritors. What combination of fractions gives a similar puzzle now?
    3 kids 1/8, 1/3, 1/2
    4 kids 1/24, 1/12, 1/3, 1/2 or
    4 kids 1/24, 1/6, 1/4, 1/2 or
    4 kids 1/12, 1/8, 1/4, 1/2
    5 kids 1/12, 1/8, 1/6, 1/4, 1/3
  3. One collector of rare pottery had 8 people to share the collection between. But one pot got broken.
    What size of collection and what proportions give a similar puzzle?
    60 pots, one broken, divided in proportions 1/60, 1/30, 1/20, 1/12, 1/10, 1/6, 1/5 and 1/3.
  4. 60 has 34 subsets of its divisors that sum to 60. It is pseudoperfect in 34 different ways.
    1. What are the divisors of 60? 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
    2. Which subsets of divisors sum to 60? For example, the smallest subset is {10,20,30}.
      Length 3:
      10, 20, 30
      Length 4:
      3, 12, 15, 30
      4, 6, 20, 30
      5, 10, 15, 30
      1, 2, 12, 15, 30
      1, 3, 6, 20, 30
      1, 4, 5, 20, 30
      Length 5:
      1, 4, 10, 15, 30
      2, 3, 5, 20, 30
      2, 3, 10, 15, 30
      2, 6, 10, 12, 30
      3, 5, 10, 12, 30
      3, 10, 12, 15, 20
      4, 5, 6, 15, 30
      Length 6:
      1, 2, 3, 4, 20, 30
      1, 2, 5, 10, 12, 30
      1, 2, 10, 12, 15, 20
      1, 3, 4, 10, 12, 30
      1, 3, 5, 6, 15, 30
      2, 3, 4, 6, 15, 30
      2, 5, 6, 12, 15, 20
      3, 4, 5, 6, 12, 30
      3, 4, 6, 12, 15, 20
      4, 5, 6, 10, 15, 20
      Length 7:
      1, 2, 3, 4, 5, 15, 30
      1, 2, 4, 5, 6, 12, 30
      1, 2, 4, 6, 12, 15, 20
      1, 3, 4, 5, 12, 15, 20
      1, 3, 5, 6, 10, 15, 20
      2, 3, 4, 5, 6, 10, 30
      2, 3, 4, 6, 10, 15, 20
      3, 4, 5, 6, 10, 12, 20
      Length 8:
      1, 2, 3, 4, 5, 10, 15, 20
      1, 2, 4, 5, 6, 10, 12, 20
    3. In how many ways is 120 pseudoperfect? 278 subsets of its divisors sum to 120.
  5. By contrast, 20 and 28 are pseudoperfect in only one way. What are those subsets?
    The divisors of 20 are 1, 2, 4, 5, 10, 20
    Its unique subset totalling 20 is 1, 4, 5, 10
    The divisors of 28 are 1, 2, 4, 7, 14, 28
  6. What is special about the pseudoperfect numbers 6 and 28?
    Both are pseudoperfect in only one way but for both these numbers the subset of divisors includes every divisor except the number itself.
    Such numbers are called Perfect numbers. The list of these is sparse and grows rapidly. It begins:
    6, 28, 496, 8128, 33550336, ... A000396
    All in this list are even and no one knows if there is an odd perfect number.

EFs for other integers

There are EFs for 2 and for 3:
Denominators of an EF for 2: 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 24
Denominators of an EF for 3: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 34, 100, 11934, 14536368
Can you find an EF for 4? for 5?

Two more unsolved problems

Here are two problems about numbers which can be written using just 3 different unit fractions.

Egyptian fractions for 4/n and the Erdös-Straus Conjecture

Although many fractions of the form 4/n can be written as a sum of just two unit fractions, others, such as 4/5 and 4/13 need three or more.

In 1948, the famous mathematician Paul Erdös (1913-1996) together with E. G. Straus suggested the following:

The Erdös-Straus Conjecture:
Every fraction 4/n can be written as a sum of three unit fractions.
It has been verified that 3 unit fractions can found for all values of n up to 1014 but as yet no one has proved it true for all values of n nor has anyone found a number n for which it is not true.
The Calculator above shows that for any given n there are many ways to choose the whole numbers, x, y and z for the three unit fraction denominators.
Using the calculator above, can you find patterns for some values of n, x, y and z?
For instance: among all the result of three fractions summing to 4/n when n is even, we have:
nxyz
63412
84520
105630
126742
...
How would you write this pattern mathematically?
4=2=1+1+1
2nnnn + 1n(n + 1)

Here is a list of all the 3-term Egyptian fractions for 4/n for n from 5 to 15.

4/5=1/2 + 1/4 + 1/20
1/2 + 1/5 + 1/10
4/6=1/2 + 1/7 + 1/42
1/2 + 1/8 + 1/24
1/2 + 1/9 + 1/18
1/2 + 1/10 + 1/15
1/3 + 1/4 + 1/12
4/7=1/2 + 1/15 + 1/210
1/2 + 1/16 + 1/112
1/2 + 1/18 + 1/63
1/2 + 1/21 + 1/42
1/3 + 1/6 + 1/14
4/8=1/3 + 1/7 + 1/42
1/3 + 1/8 + 1/24
1/3 + 1/9 + 1/18
1/3 + 1/10 + 1/15
1/4 + 1/5 + 1/20
1/4 + 1/6 + 1/12
4/9=1/3 + 1/10 + 1/90
1/3 + 1/12 + 1/36
1/4 + 1/6 + 1/36
1/4 + 1/9 + 1/12
4/10=1/3 + 1/16 + 1/240
1/3 + 1/18 + 1/90
1/3 + 1/20 + 1/60
1/3 + 1/24 + 1/40
1/4 + 1/7 + 1/140
1/4 + 1/8 + 1/40
1/4 + 1/10 + 1/20
1/4 + 1/12 + 1/15
1/5 + 1/6 + 1/30
4/11=1/3 + 1/34 + 1/1122
1/3 + 1/36 + 1/396
1/3 + 1/42 + 1/154
1/3 + 1/44 + 1/132
1/4 + 1/9 + 1/396
1/4 + 1/11 + 1/44
1/4 + 1/12 + 1/33
4/12=1/4 + 1/13 + 1/156
1/4 + 1/14 + 1/84
1/4 + 1/15 + 1/60
1/4 + 1/16 + 1/48
1/4 + 1/18 + 1/36
1/4 + 1/20 + 1/30
1/4 + 1/21 + 1/28
1/5 + 1/8 + 1/120
1/5 + 1/9 + 1/45
1/5 + 1/10 + 1/30
1/5 + 1/12 + 1/20
1/6 + 1/7 + 1/42
1/6 + 1/8 + 1/24
1/6 + 1/9 + 1/18
1/6 + 1/10 + 1/15
4/13=1/4 + 1/18 + 1/468
1/4 + 1/20 + 1/130
1/4 + 1/26 + 1/52
1/5 + 1/10 + 1/130
4/14=1/4 + 1/29 + 1/812
1/4 + 1/30 + 1/420
1/4 + 1/32 + 1/224
1/4 + 1/35 + 1/140
1/4 + 1/36 + 1/126
1/4 + 1/42 + 1/84
1/4 + 1/44 + 1/77
1/5 + 1/12 + 1/420
1/5 + 1/14 + 1/70
1/5 + 1/20 + 1/28
1/6 + 1/9 + 1/126
1/6 + 1/12 + 1/28
1/6 + 1/14 + 1/21
1/7 + 1/8 + 1/56
4/15=1/4 + 1/61 + 1/3660
1/4 + 1/62 + 1/1860
1/4 + 1/63 + 1/1260
1/4 + 1/64 + 1/960
1/4 + 1/65 + 1/780
1/4 + 1/66 + 1/660
1/4 + 1/68 + 1/510
1/4 + 1/69 + 1/460
1/4 + 1/70 + 1/420
1/4 + 1/72 + 1/360
1/4 + 1/75 + 1/300
1/4 + 1/76 + 1/285
1/4 + 1/78 + 1/260
1/4 + 1/80 + 1/240
1/4 + 1/84 + 1/210
1/4 + 1/85 + 1/204
1/4 + 1/90 + 1/180
1/4 + 1/96 + 1/160
1/4 + 1/100 + 1/150
1/4 + 1/105 + 1/140
1/4 + 1/108 + 1/135
1/4 + 1/110 + 1/132
1/5 + 1/16 + 1/240
1/5 + 1/18 + 1/90
1/5 + 1/20 + 1/60
1/5 + 1/24 + 1/40
1/6 + 1/11 + 1/110
1/6 + 1/12 + 1/60
1/6 + 1/14 + 1/35
1/6 + 1/15 + 1/30
1/7 + 1/10 + 1/42
1/8 + 1/10 + 1/24
1/9 + 1/10 + 1/18
Can you spot any further patterns here?
Use the Calculator above to help with your investigations.

If you do find any more, let me know (see contact details at the foot of this page) and I will put your results here.
If we can find a set of cases that cover all values of n, then we have a proof of the Erdös-Straus conjecture.

Here are some simple cases for 4/n that we can easily see have 3 unit fractions since the 3 fractions need not have different denominators:
Continuing in this way, we can eliminate many forms of denominator from the list of those needing verification - but no one has managed to find a proof for all n yet!

In Mordell's Diophantine Equations published by Academic Press in 1969, he showed that we can reduce the unknown (unproven) cases to just those prime denominators n which have a remainder of 1, 112, 132, 172, 192 or 232 when divided by 840 (see A139665).

Things to do /

  1. The number of solutions to 4/n as a sum of 3 unit fractions is:
    The first value is 4/3:
    1 solution for 4/3 = 1/1+1/4+1/12
    1 solution for 4/4 = 1/2+1/3+1/6
    2 ways for 4/5
    5 ways for 4/6 = 2/3
    ...
    The series of counts is (0,0), 1, 1, 2, 5, ...
    How does it continue?
    Check your answers with A073101 in Neil Sloane's Online Encyclopedia of Integer Sequences.
    If the Erdös-Straus Conjecture is true then the only zeroes in the whole infinite series are for n=1 and 2.

    With thanks to Robert David Acker, Jr. for suggesting this topic.

5/n = 1/x + 1/y + 1/z?

Another famous mathematician, Sierpinski suggested in 1956 that the same applied to all fractions of the form 5/n, that is that each of these also can be expressed as a sum of 3 unit fractions.

There are:
0 solutions for 5/2
1 solution for 5/3: 5/3=1/1+1/2+1/6
2 for 5/4: 5/4=1/1+1/5+1/20 and 1/1+1/6+1/12;
1 for 5/5; what is it?
The number of solutions this time is the series 0,1,2,1,1,3,5,9,6,3,12,... which is A075248 in Neil Sloane's Online Encyclopedia of Integer Sequences. If the conjecture is true, then there are no zeroes in this series apart from the starting value.

Things to do /

    1. Find the single set of 3 unit fractions with a sum of 5/6.
    2. Find the three sets of 3 for 5/7.
    3. What formulae can you find for special cases of 5/n as a sum of 3 unit fractions?
  1. From the table of lengths of the shortest Egyptian fractions above, find a fraction that needs 5 unit fractions for its Egyptian fraction.
  2. Can you find a fraction that cannot be written using less than 6 unit fractions for its Egyptian fraction?
  3. Investigate Egyptian fractions which have only odd denominators.
    Is it possible to find a sum of odd Egyptian fractions for every fraction a/b?

The above will give you some ideas for your own experiments and the References below point to more information and ideas.
Happy calculating!

Smallest Denominators

Apart from the shortest Egyptian fractions (those with the fewest unit fractions), we can also look for the smallest numbers in the denominators.
As we saw at the start of the Fixed Length Egyptian Fractions section above, the smallest denominators do not always appear in the shortest Egyptian fractions.

The 2/n table of the Rhind Papyrus

Here is the Table at the start of the Rhind mathematical papyrus. It is a table of unit fractions for 2/n for the odd values of n from 3 to 101.
Sometimes the shortest Egyptian fraction is ignored in the table in favour of a longer decomposition. Only one sum of unit fractions is given when several are possible. The scribe tends to favour unit fractions with even denominators, since this makes their use in multiplication and division easier. The Egyptian multiplication method was based on doubling and adding, in exactly the same way that a binary computer uses today, so it is easy to double when the unit fractions are even.
Also, he prefers to use smaller numbers. Their method of writing numerals was decimal more like the Roman numerals than our decimal place system though. He seems to reject any form that would need a numeral bigger than 999. All the shortest forms and alternative shortest forms are given here in an extra column.
2 = 1 + 1 + 1 + 1
nabcd

nabcdshortest?
5315   
7428   
9618  5 45
11666   
13852104 ×7 91
151030  8 120
9 45
12 20
17125168 ×9 153
191276114 ×10 190
211442  11 231
12 84
15 35
2312276   
251575  13 325
271854  15 135
24 378
292458174232×15 435
3120124155 ×16 496
332266  21 77
18 198
17 561
353042  21 105
20 140
18 630
3724111296 ×19 703
392678  24 104
21 273
20 780
4124246328 ×21 861
434286129301×22 946
453090  36 60
35 63
27 135
25 225
24 360
23 1035
nabcdshortest?
4730141470 24 1128
4928196  25 1225
5134102  30 170
27 459
26 1326
5330318795 ×27 1431
5530330  40 88
33 165
28 1540
5738114  33 209
30 570
29 1653
5936236531 ×30 1770
6140244488610×31 1891
6342126  56 72
45 105
36 252
35 315
33 693
32 2016
6539195  45 117
35 455
33 2145
6740335536 ×34 2278
6946138  39 299
36 828
35 2415
7140568710 ×36 2556
7360219292365×37 2701
7550150  60 100
45 225
42 350
40 600
39 975
38 2850
nabcdshortest?
7744308  63 99
42 462
39 3003
7960237316790×40 3160
8154162  45 405
42 1134
41 3321
8360332415498×42 3486
also
166 249 498
8551255  55 187
45 765
43 3655
8758174  48 464
45 1305
44 3828
8960356534890×45 4005
184 of length 3
9170130  52 364
49 637
46 4186
9362186  51 527
48 1488
47 4371
9560380570 ×60 228 
57 285
50 950
48 4560
9756679776 ×49 4753
9966198  90 110
63 231
55 495
54 594
51 1683
50 4950
101202303606 ×51 5151
All those with n a multiple of 3 follow the same pattern:
2/3n = 1/2n + 1/6n
But there are still some mysteries here.
For instance why choose
2/95 = 1/60 + 1/380 + 1/570
instead of the much simpler
2/95 = 1/60 + 1/228?
Why stop at 103? There is a sum for 2/103 with two unit fractions but it contains a four digit number:
2/103 = 1/52 + 1/5356
and all of the other 65 of length 3 contain a denominator of at least 1236.
The one with this least maximum denominator is: 2/103 = 1/60 + 1/515 + 1/1236
There are only two of length 4 that don't use four digit numbers:
2/103 = 1/103 + 1/206 + 1/309 + 1/618
2/103 = 1/72 + 1/309 + 1/824 + 1/927.
The Calculator above may help with your investigations.

Things to do /

  1. Is there a pattern common to all the 2/5n forms in the papyrus table?
  2. Is there a pattern common to all the 2/7n forms in the papyrus table?
  3. Which fractions in the table could be found by the Fibonacci method?

Factored Egyptian Fractions

Some fractions have a special form of EF where each denominator is a factor of the next. For example:
4 = 1 + 1 + 1
522×22×2×5
In fact, these are useful for a system of measurements such as the pounds-shillings-pence monetary system or ounces-pounds-stones system of weights.
There were 12 pence in every shilling, 20 shillings in every pound so £2 3s 11p could be represented as £ 2 + 3/20 + 11/(20×12).
Similarly there are 16 ounces in 1 pound weight, 14 pounds in one stone. 2 stone 9 pounds and 3 ounces is 2 + 9/14 + 3/(14×16) ounces.

Every fraction has a Factored EF

In 1973 Robert Cohen proved that every fraction has such a form and that only when a number is irrational (that is, not a fraction, such as √2 or π) does such a series have an infinite number of terms.
Every fraction has a factored Egyptian Fraction form
Cohen only focuses on a "greedy" method of generating them and restricts his attention to those where each new factor is never smaller than the one before it when we order the denominators in increasing size. With this restriction, he proves a factored Egyptian fraction always exists and that it is unique.
3 =  1 + 1  = 1 + 1
833×844×2
factors 3,8
increasing
factors 4,2
decreasing

Cohen called these Egyptian fraction expansions but since 1973 much has been written on Egyptian fractions so, to avoid confusion, we will use the term factored Egyptian Fractions here. We will not have Cohen's restriction about a new factor never being smaller than its predecessor; the new factor introduced in the next denominator (when the denominators are in increasing order) may be bigger than the previous one or the same or smaller here.

We will find that it is sometimes more convenient to look at the list of successive factors of a factored EF, for example

13/20 = 1/2 + 1/8 + 1/40 = 1/2 + 1/(2×4) + 1/(2×4×5) has the successive factors 2, 4, 5
7/8 = 1/2 + 1/4 + 1/8 = 1/2 + 1/(2×2) + 1/(2×2×2) has the successive factors 2, 2, 2

Calculator A Factored Egyptian Fraction Calculator

Factored EF C A L C U L A T O R

length
factored EFs for
--
with successive factors
R E S U L T S

calculator: Fraction ↔ EF Shortest EF Fixed Length EFs Factored EFs Harmonic Numbers

Lengths of shortest Factored EFs

Every unit fraction has a factored Egyptian Fraction form:
1/2 = 1/3 + 1/6
1/3 = 1/4 + 1/12
1/4 = 1/5 + 1/20
1/5 = 1/6 + 1/30
1/6 = 1/7 + 1/42
1/7 = 1/8 + 1/56
which is summarised by the algebraic identity
1=1+1 Factored expansion equation
nn + 1n ( n + 1)
A similar identity holds for all fractions 2/odd:
2=1+1
2n + 1n + 1(n + 1) ( 2n + 1)
Here is a table of the lengths of the shortest factored EFs for fractions less and one and with a denominator up to 30, similar to the one above for all EFs. This table includes all fractions whether in their lowest form or not:
Length of the Shortest Factored EF for Fraction T/B
KEY: . means the fraction t/b is not less than 1.
12 is the minimum number of unit fractions that are needed to sum to t/b.
Find T (top or numerator) down the side and B (bottom or denominator) across the top.
\B:                 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3
T\  2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
 1| 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 2| . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 3| . . 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2
 4| . . . 3 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 3 2
 5| . . . . 3 4 2 2 2 3 2 3 2 2 2 3 2 2 2 3 3 3 2 2 3 2 2 2 2
 6| . . . . . 5 2 2 2 2 2 4 3 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2
 7| . . . . . . 3 3 3 3 2 2 2 3 3 4 2 3 2 2 3 3 2 3 2 2 2 3 2
 8| . . . . . . . 4 3 4 2 4 2 2 2 5 2 3 2 2 2 2 2 3 3 3 2 3 2
 9| . . . . . . . . 4 4 2 4 3 2 2 2 2 4 3 3 4 3 2 3 2 2 3 4 2
10| . . . . . . . . . 5 3 3 4 2 2 3 2 2 2 4 3 4 2 2 3 2 2 2 2
11| . . . . . . . . . . 4 5 3 4 3 4 3 4 2 2 2 5 3 4 3 3 3 3 2
12| . . . . . . . . . . . 6 5 3 2 5 2 3 2 2 2 2 2 5 4 2 3 4 2
13| . . . . . . . . . . . . 6 5 3 3 3 4 3 4 3 3 2 2 2 3 4 3 3
14| . . . . . . . . . . . . . 6 3 5 3 5 3 2 3 4 2 3 2 2 2 4 3
15| . . . . . . . . . . . . . . 4 4 3 5 2 4 3 4 2 2 3 2 2 2 2
16| . . . . . . . . . . . . . . . 5 4 5 3 3 4 4 2 4 4 3 2 3 2
17| . . . . . . . . . . . . . . . . 5 6 4 5 3 6 3 4 4 3 3 3 3
18| . . . . . . . . . . . . . . . . . 7 4 5 4 4 2 5 4 2 3 4 2
19| . . . . . . . . . . . . . . . . . . 5 6 5 5 3 3 5 4 3 5 3
20| . . . . . . . . . . . . . . . . . . . 7 5 7 3 3 3 4 4 4 2
21| . . . . . . . . . . . . . . . . . . . . 6 6 3 5 5 3 2 4 3
22| . . . . . . . . . . . . . . . . . . . . . 7 4 4 5 4 3 3 4
23| . . . . . . . . . . . . . . . . . . . . . . 5 6 4 5 4 4 3
24| . . . . . . . . . . . . . . . . . . . . . . . 7 6 4 5 6 3
25| . . . . . . . . . . . . . . . . . . . . . . . . 7 6 4 5 3
26| . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 5 5
27| . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 4
28| . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6
29| . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Here is the table of the same fractions but this time counting the number of factored EFs there are of the shortest length:
Number of Shortest Factored EFs for Fraction T/B
\B:                  1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3
T\   2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
 1 | 1 1 2 1 3 1 3 2 3 1 5 1 3 3 4 1 5 1 5 3 3 1 7 2 3 3 5 1 7
 2 | . 1 1 1 1 1 2 2 1 1 3 1 1 3 3 1 2 1 3 3 1 1 5 2 1 3 3 1 3
 3 | . . 1 1 1 1 2 1 2 1 2 1 2 1 2 1 3 2 3 1 2 1 3 1 2 2 2 1 3
 4 | . . . 1 1 1 1 1 1 1 1 1 1 2 2 1 2 1 1 2 1 1 3 2 1 2 1 2 3
 5 | . . . . 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 3 3 2 2 1 3 1 2 1 3
 6 | . . . . . 1 1 1 1 1 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 1
 7 | . . . . . . 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1
 8 | . . . . . . . 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2
 9 | . . . . . . . . 1 1 1 2 1 1 1 1 1 1 1 1 4 1 2 2 1 1 1 2 2
10 | . . . . . . . . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1
11 | . . . . . . . . . . 1 2 1 1 2 1 2 2 1 1 1 1 2 2 1 2 3 1 1
12 | . . . . . . . . . . . 2 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1
13 | . . . . . . . . . . . . 1 1 1 1 1 1 2 3 1 1 1 1 1 1 3 1 2
14 | . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1
15 | . . . . . . . . . . . . . . 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1
16 | . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 2 1 1 1 1 1
17 | . . . . . . . . . . . . . . . . 1 2 1 3 1 1 1 2 1 1 2 1 3
18 | . . . . . . . . . . . . . . . . . 2 1 1 1 1 1 2 2 1 1 1 1
19 | . . . . . . . . . . . . . . . . . . 1 3 1 1 1 1 2 2 1 2 2
20 | . . . . . . . . . . . . . . . . . . . 3 1 1 1 1 1 1 1 1 1
21 | . . . . . . . . . . . . . . . . . . . . 1 1 1 2 1 1 1 1 1
22 | . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 1 1 1 1
23 | . . . . . . . . . . . . . . . . . . . . . . 1 2 1 2 1 1 1
24 | . . . . . . . . . . . . . . . . . . . . . . . 2 2 1 1 2 1
25 | . . . . . . . . . . . . . . . . . . . . . . . . 2 2 1 1 1
26 | . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1 1
27 | . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1
28 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
29 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Are there an infinite number of factored EFs for every fraction?

Since every EF has a factored EF, we can try the method we used above to show that every factored EF can be expanded.
If the final factor was n (the ratio between the final two unit fractions in a factored EF), then we can replace it using the Expansion Equation that we used earlier, as follows:
t = ... 1+1 = ...1+1 + 1
ban aa(n+1) an (n + 1) a
This is easier when we look at the list of successive factors so that, for example
13/20 = 1/2 + 1/8 + 1/40 = 1/2(1 + 1/4(1 + 1/5)) = 1/2 + 1/(2×4) + 1/(2×4×5) has the successive factors 2, 4, 5.
The final factor n of any factored EF can be replaced by the two factors n + 1, n:
t/bfactored EFsuccessive
factors
expanded
factors
expanded
factored EF
13/201/2 + 1/8 + 1/40 2, 4, 52, 4, 6, 51/2 + 1/8 + 1/48 + 1/240
1/2 + 1/8 + 1/48 + 1/2402, 4, 6, 52, 4, 6, 6, 51/2 + 1/8 + 1/48 + 1/288 + 1/1440
So every fraction has a factored EF as we saw above and we have just shown that we can extend each one by one more factored term as often as we like.

The Harmonic Numbers

Summing the reciprocals from 1 up to n and the series of such sums have long intrigued mathematicians. The sums of the reciprocals of 1 up to n are called the Harmonic Numbers H(n) and the series H(1), H(2), H(3), ... is called the Harmonic Series:
H(n) = 1 +  1 + 1 + 1   + ... +  1
1234n
The series is called Harmonic because if we stretch a string tightly and twang it, we hear a certain note. Stopping the string at the half way point makes a sound an octave above the first note. If instead we take just one third of the length we get another note that seems harmonious to the ear in relation to the whole string. We also get harmonious sounds if we take one quarter and one fifth and so on for some time.
Pythagoras first noted this connection with harmonious sound and the lengths of plucked strings.
When a string is plucked as on a violin for instance, there is not only the "pure" note but also other quieter notes produced by these "harmonic" notes called overtones.

The first few values of H(n) are:

n 1 2 3 4 5 6 7 ...
H(n) 1 3 11 25 137 49 363 A001008the Wolstenholme numbers
1 2 6 12 60 20 140 A002805

n 1 2 3 4 5 6 7 ...
H(n) 1 3 11 50 274 1764 13068 A000254 the Stirling numbers of the first kind
1 2 2×3 2×3×4 2×3×4×5 2×3×4×5×6 2×3×4×5×6×7 A000142 the Factorial numbers

The second table is the same fractions for H(n) but without simplifying the sums.
The numerators give the total number of cycles in all permutations of n+1 and the fractions give the ratio (probability) of a random permutation on n+1 letters having exactly two cycles!
What can we say about H(n)? Is any Harmonic number ever exactly an integer for instance?
Subtracting one Harmonic number from another larger one H(n) – H(k) gives us the sum of a consecutive set of the unit fractions 1/(k+1) + ... + 1/n. Are any of these every integers?
Unfortunately, the answers are no; no finite sum of the series of unit fraction starting at 1 or at another unit fraction will ever sum to a whole number.

The sum of all unit fractions

The series of reciprocals of all the natural numbers from 2 onwards is called the Harmonic Series and the question of whether or not its sum has a finite value or not is now a classic maths problem. There is a very easy proof to show that the harmonic series series "diverges", that is, summed for ever, it gets infinitely large.
In case you want to think about it yourself, the answer is revealed in an optional section:
Let's look at the first term: 1/2,
The next 2 terms: 1/3 + 1/4 but 1/3 > 1/4 so
1+1 1+1=1
34442

Now look at
the next 4 terms:
1+1+1 +1 1+1+1+1 = 1
567888882

similarly the sum of
the next 8 terms will exceed 1/2.
Since the Harmonic series sum goes on for ever, then we can always find another batch of 2n terms whose sum adds more than 1/2 to the total,
so the total is always larger than any given number, it never settles down to a fixed value, it grows for ever or "diverges".
This result has been known since at least 1650 (Pietro Mengoli).

The Overhanging books puzzle

bookstack There is a surprising application of the divergence of the Harmonic Series that makes a good Science Fair demonstration.
Suppose we have a shelf of identical books (or bricks or dominoes etc).
If we lay them down one on top of another what is the maximum overhang we can achieve?
Can the top book ever completely overhang the bottom book?

bookstack For two books we can get an overhang of 1/2 a book length;
The two books now have a centre of gravity in the centre of their overlap, so the bottom one can overhang and extra 1/4 of a book length.
The centre of gravity of the three books means they can have an overhang of 1/6, and so on for more books.
The extra overhang for four and more books is 1/8, 1/10, 1/12, ... .
The total overhang for n books is therefore H(n-1)/2.

How many books will it take before the overhang exceeds the length of one book?
Answer: H(4)/2 = 25/24 which is bigger than 1 so:

5 stacked books are sufficient for the top one to totally overhang the bottom one!

Can we get a bigger overhang? Yes! Since the Harmonic series diverges, we can get an overhang as large as we like, but it may take a large number of books and very delicate balancing! Since it is very difficult to find lots of identically sized books, try playing cards.
Theoretically for an overhang of 2 book lengths, we need to find the value of n for which H(n) > 4
H(30) = 3.99498713092
H(31) = 4.02724519544
so we need 32 books to get an overhang of 2 complete books!
For 3 book lengths, the first n with H(n) > 6 is n = 227 so that stack would be 228 books tall and n = 1674 before H(n) exceeds 8.

Calculator A Harmonic Number Calculator

The values of the Harmonic function in this calculator are computed approximately.
All digits shown are correct and are within the accuracy of the approximation.
Harmonic Number C A L C U L A T O R
H(  )


calculator: Fraction ↔ EF Shortest EF Fixed Length EFs Factored EFs Harmonic Numbers
If we had 1,000,000 playing cards perfectly arranged to give the maximum overhang of the top card, how many card lengths would it overhang?
H(1 000 000) = 14.392726722865724 so the top card would overhang by more than 7 card lengths.

EF Links and References

The following two books are recommended if you want to read more about the extraordinary Hungarian mathematician Paul Erdös On the History of Egyptian mathematics, I recommend:
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