# 2021

2021 - some of the more unusual and unique fun facts about the number.
Some are left as puzzles for you to solve with a button to show the solutions.
Many of the facts below are derived from Neil Sloane's On-line Encyclopedia of Integer Sequences where links to the sequences containing 2021 listed here are shown as, for instance,
A000045 ( the Fibonacci numbers).
The icon means there is a You Do The Maths section of questions to start your own investigations.

## The digits

### 2021 using 1 to 10

Alex Bellos in his Monday Puzzle for 28 December 2020 in the Guardian newspaper posed the problem of using the numbers 10 down to 1, in that order , together with the four simple maths operations of +, −, ×, ÷ and brackets to make 2021. There are some solutions in his Answer page.
One inadmissible version from Chris Smith is
10 + (9 + 8 + 7 + 6 )(5 + 43 − 2) + 1
which is inadmissible because it uses the "to-the-power-of' operation in 43.

#### You Do The Maths...

Please email me with any answers you find here so that they may be included with your name.
1. Using only those 4 arithmetic operations and brackets, can you make 2021 by inserting them in between the numbers 10 9 8 7 6 5 4 3 2 1?
10 − 9 × 8 × 7 × 6 + 5 − 4 × 3 × 2 − 1
2. What if you were allowed to use the same 10 numbers but in any order?

### Two successive numbers put together

2021 is of the form 20 then 21 or n followed by n+1 when written in base 10.
The next year like this is 2122, in 101 year's time.
A001704

2021 is also composed just of the digits 0,1 and 2.

#### You Do The Maths...

1. It is easy to write the series of numbers which are just 2 successive numbers joined together such as 2021.
Suppose we allowed any number of consecutive integers to be joined together, such as 5678 and 345.
Write down the full list of such numbers in order up to 2021.
How many are there?
The list is:
12, 23, 34, 45, 56, 67, 78, 89, 123, 234,
345, 456, 567, 678, 789, 910, 1011, 1112, 1213, 1234,
1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021
so there are 27 up to 2021.
A035333
2. Up to 2021, how many years are there whose largest digit is 2?
2, 12, 20, 21, 22, 102, 112, 120, 121, 122,
200, 201, 202, 210, 211, 212, 220, 221, 222, 1002,
1012, 1020, 1021, 1022, 1102, 1112, 1120, 1121, 1122, 1200,
1201, 1202, 1210, 1211, 1212, 1220, 1221, 1222, 2000, 2001,
2002, 2010, 2011, 2012, 2020, 2021, ...
There are 45 such numbers less than 2021, 2021 is the 46th.
A277964
3. Suppose now that we want all the numbers up to 2021 that we can write using some of all of the digits 0,1 and 2.
There is a quick way to find out how many are less than 2021. How many are there and what is the quick method?
Such numbers are just the integers written in base 3.
If the first is 0, the second 1 and so on then 2021 as a base 3 number is
2×33 + 0×32 + 2×31 + 1 = 61
2021 is the 62nd number.

### Squares and reversals

The square of 2021 is 4084441.
The reverse of 2021 is 1202 whose square is 1444804, which is just the reverse of 20212
Such numbers are not very common ...

#### You Do The Maths...

1. How many numbers can you find whose squares are reversals of each other, up to 1202 (the smaller of 2021 and its reversal)?
12, 13, 33, 102, 103, 112, 113, 122, 1002, 1003,
1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103, 1112, 1113,
1121, 1122, 1202, ...
There are 23 including 1202.
A106323

### Palindromes

2021 in base 46 is the two-digit palindrome {43, 43} since
2021 = 43×46 + 43

Also 2021 times its reversal 1202 is the palindromic number 2429242.
A106323

#### You Do The Maths...

1. How many other numbers smaller than 2021 can you find that make a palindrome when they are multiplied by their reversal?
For example, 211×112 is 23632, a palindrome but this is not the smallest example. In your list include both the number and its reversal, so 112 and 211 in this example. Show the answer
The complete list up to 2021 is:
1, 2, 3, 11, 12, 21, 22, 101, 102, 111,
112, 121, 122, 201, 202, 211, 212, 221, 1001, 1002,
1011, 1012, 1021, 1022, 1101, 1102, 1111, 1112, 1121, 1201,
1202, 1211, 2001, 2002, 2011, 2012, 2021, ...
There are 36 smaller numbers.
A062936

### 2021 and pi, e and phi

If we list the decimal digits of π, e and the golden ratio φ we have the following as their first 20 digits:
 # π = e = φ = 1 . 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 5 2 . 7 1 8 2 8 1 8 2 8 4 5 9 0 4 5 2 3 5 4 1 . 6 1 8 0 3 3 9 8 8 7 4 9 8 9 4 8 4 8 2
The 13th digits of all of them is the digit 9 as shown.
There are only 19 other places where all three decimal expansions agree in the first 2020 digits and then at the 2021st, in all three, we find the digit 3.

A266002
If we try to include square-roots also, the smallest is √46 which has a 3 at place 2021.

### Runsums

A Runsum is a sum of a run of consecutive integers:
2021 = 20 + 21 + ... + 65 + 66
2021 = 26 + 27 + ... + 67 + 68
2021 = 1010 + 1011

The sum of all the numbers up to 2021 is the runsum from 1 to 2021 and is 2043231.

2021 is also the link between two runsums, making the "friendly runsum":
195 + 196 + ... + 2020 + 2021 = 2021 + 2022 + ... + 2850 + 2851 = 2024316

### Primes

2021 is not prime but 2021 = 43×47, a product of two primes called a semi-prime. The next such number is 2491.
A006094

#### 2021 in a non-decimal base

In base 3, 20213 is 61 and 61 is prime.

#### You Do The Maths...

1. When was the last year that was a semi-prime number? Show the answer
2019 = 3 × 673
2. Find all the bases up to base 10 in which 2021 is a prime number.
20213 = 61
20214 = 137
20217 = 701
are the only primes.

## Pythagorean Triangles A Pythagorean Triangle is a right-angled triangle whose sides are whole numbers, such as 3, 4, 5.
Once we have one such triangle, we can always multiply all the sides by 2 say to get 6-8-10 or by 3 to find 9-12-15 and so on. All these have the same shape.
If we want different shapes then the smallest Pythagorean triangle of each shape is called a primitive Pythagorean triangle or PPT for short. All Pythagorean triangles are PPTs or a multiple of a PPT.
There are 4 Pythagorean triangles with a side of length 2021:
43×(47,1104,1105) = 2021, 47472, 47515;
47×(43,924,925) = 2021, 43428, 43475;
2021,2042220,2042221;
180,2021,2029
No Pythagorean triangle has a hypotenuse (longest side) of 2021.
No Pythagorean triangle has a perimeter (sum of all three sides) of 2021.
No Pythagorean triangle has an area of 2021.
A009000: the hypotenuses
A009096: the perimeters
A055112: the areas

## Other number facts

### 2021!

The product of all the numbers from 1 to 2021 is called "the factorial of 2021" and written 2021! .
2021! has 5805 digits. It starts 780301939..... but how does it end?
With a 0 because it contains 10 as a factor.
But 100 and 1000 are also factors so that gives at least 1+2+3=6 zeroes at the end.
In fact there are many more zeroes at the end of 2021! .
How many zeroes are at the end of 2021! ?
The answer is in the You Do The Maths... section below.

### Remainders

Here is a table of the remainders r when 2021 is divided by 2 up to 10:
 d= r= 2 3 4 5 6 7 8 9 10 1 2 1 1 5 5 5 5 1
The remainder is 5 when divided by each of 6, 7, 8 and 9.
When is the next? See You Do The Maths... section below.

### Powers

The nearest power to 2021 is 452 = 2025.
2021 is close to 215/2 = 212 √21 = 2022.92 but in two years time it is closer still!

### You Do The Maths...

1. Find an easy way to compute the number of 0s at the end of 2021 .
Each 0 corresponds to a multiple of 10 in the product.
10 = 5 × 2.
There are far more even numbers in the product than multiples of 5, so we can always match each 0 at the end with a factor 5 (and a factor 2) in the product.
 2021/5 = 404.2 so there are 404 numbers with a factor 5 2021/52 = 80.84 so there are 80 numbers with an extra 5 as a factor 2021/53 = 16.168 so there are 16 with another 5 as factor 2021/54 = 3.2336 so there are just 5 with one more factor 5 total = 503
35 is 3125 which is bigger than 2021 so that accounts for all the 5 factors in 2021!
2. What formula will give the number of digits in 2021! ?
Let x be Log10 ( 2021! ) then 10x is 2021! .
When rounded up, this gives the number of decimal digits in 2021! (since this is not an exact power of 10)
(Log10 is 2.00432 and Log10 is 2.99957. All the values of the logs to base 10 of the numbers from 100 to 999 begin with 2 and all contain 3 digits.

Find out more about logs and how to use them on the Formula for Fibonacci Numbers page

3. Which was the first year to have a remainder of 5 when divided by 6 and 7 and 8 and 9? (Easy!)
How does the list continue up to 2021? (Harder)
When is the next year with this property?
The first year was year 5.
If we add 6 to any year with remainder 5 when divided by 6, then it also will have the same remainder (mod 6).
If we want multiples of both 6 and 7 to have remainder 5, we can add 6×7 = 42.
The Lowest Common Multiple of 5, 6, 7 and 8 is 504.
Any multiple of 504 can be added to 5 to produce a number with remainder 5 when divided by each of 6, 7, 8 and 9:
The list continues... 5+504=509, 5+2×504=1013, 1517, 2021, 2525
4. Make a list of all the whole numbers that are a perfect power, that is, are of the form ab for whole numbers a and b, up to 400.
Up to 400 they are:
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400
A001597
5. How do you find numbers close to 2021 of the form bp with the power p is a "simple" fraction?
© 2021 Created: 31 December 2020 Dr Ron Knott 