# Interesting facts about the numbers 1 - 31

I'm a maths teacher and want to create a calendar for my classroom.

I'm looking to compile some interesting facts about each of the numbers 1 through to 31.

The hard part is they must be at a level where 11-18 year-olds can understand them.

Along the lines of: 2 is the smallest prime number, only even prime number.



Any help would be greatly appreciated.

• Like what is the multiplicative identity? – randomgirl Jun 16 '18 at 21:37
• Google "perfect numbers"; it'll give you somethign for $6$ and $28$. – Kaj Hansen Jun 16 '18 at 21:37
• I suppose wiki has a list of properties like that. – Kenny Lau Jun 16 '18 at 21:46
• @MathewDuxbury: See: archimedes-lab.org/numbers/Num1_69.html – Moo Jun 16 '18 at 21:47
• 17 is a fermat's prime which also happens to be the sum of the first 4 consecutive primes. – lanskey Jun 16 '18 at 22:14

## 1 Answer

Very interesting idea- I like it. How about these:

$1$ is the only number with one factor, or the only number which is the factorial of two numbers ($0!=1!=1)$

$2$ is the only even prime

$3$ is the best integer approximation of $\pi$ and $e$.

Every even square is a multiple of $4$.

$5$ is the only odd number for which every number ending in it is composite.

$6$ is the smallest perfect number

$7$ is the smallest number that cannot be represented as the sum of three integer squares (e.g. $6=2^2+1^2+1^2$). Or $\frac n7$ always features the same digits in the same order with the start varying.

$8$ is the only Fibonacci Number besides $1$ that is a perfect cube.

$9$ is roughly $2\pi+e$

$10$ is the sum of the first three primes, the first four factorials, and the first four positive integers.

The powers of $11$ can be found by Pascal's Triangle.

$12$ is the smallest sublime number, of which there are only two known.

$13$ is the smallest emirp.

$14$ is the smallest satisfier of the Shapiro Inequality.

$15$ is $1|5$ and $1+2+3+4+5$

$16$ is the smallest perfect fourth power besides $0$ and $1$

$17\approx12\sqrt2$ and so we can use it to approximate $\sqrt2$

$18$ is the first non-square number expressible as $p\cdot q^2$, with $p,q$ integers and $q>p$

Any natural number can be made by summing $19$ $4$th powers.

The product of the divisors $(1,2,4,5,10)$ with the proper divisors $(2,4,5,10)$ of $20$ is $20$.

$21$ is a triangular number.

$22\approx7\pi$, thus $\pi\approx\frac{22}{7}$ is used where a fractional approximation is needed.

$23$ is the smallest number irrepresentable as the sum of nine or fewer positive cubes.

$a=24$ is the largest and only non-trivial solution to $1^2+2^2+...+a^2=b^2$

If the last two digits of a number are a multiple of $25$, the number is a multiple of $25$.

$26$ is the only number of both the form $a^2+1$ and the form $a^3-1$.

$27$ is the only integer that is three times the sum of its digits. Or the first significant input of the Collatz Conjecture.

$28$ is the only known number which is the sum of the first $a$ primes, the first $b$ non-negative integers, and the sum of the first $c$ non-primes. (Here, $a=7, b=5, c=5$)

$29$ cannot be made using addition, subtraction, multiplication and division with $1,2,3,4$, suing each only once.

$30=(2220422932)^3+(-2218888517)^3+(-283059965)^3$, and was the first number that was significantly challenging to represent as the sum of three cubes.

$31,331,3331,33331,333331,3333331,33333331$ are all prime.