Fun With Numbers

1. Square numbers ending in 5.

For n5²: n × (n+1), then append “25”.

Example: 65² → 6 × 7 = 42, append 25 → 4225.

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2. Amicable Numbers.

Amicable numbers are pairs of numbers where each number is the sum of the other’s proper divisors (excluding the number itself).

The lowest pair of amicable numbers are:- 220 and 284.

Proper divisors of 220 – 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 – added together = 284

Proper divisors of 284 – 1, 2, 4, 71, 142 – added together = 220

The mathematical formula:-

Two different numbers a and b form an amicable pair if:

s(a)=b and s(b)=a

Amicable numbers are rare.

The next four lowest pairs are:-

(1,184, 1,210)
- Sum of proper divisors of 1,184 = 1,210
- Sum of proper divisors of 1,210 = 1,184
(2,620, 2,924)
- Sum of proper divisors of 2,620 = 2,924
- Sum of proper divisors of 2,924 = 2,620
(5,020, 5,564)
- Sum of proper divisors of 5,020 = 5,564
- Sum of proper divisors of 5,564 = 5,020
(6,232, 6,368)
- Sum of proper divisors of 6,232 = 6,368
- Sum of proper divisors of 6,368 = 6,232

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3. Multiplying by 25 Fast.

Multiply any number by 25 by dividing by 4 and shifting two decimals (e.g. 84×25 = (84/4)=21 → 2100).

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4. Negative 40 coincidence.

-40 °C equals -40 °F, the only point where Celsius and Fahrenheit meet

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5.

Leyland Numbers.

Leyland numbers are numbers of the form:

$x^y + y^x$

where $x > 1$ and $y > 1$ are integers.

Using small values of $x$ and $y$ :

$x$	$y$	Calculation	Leyland number
2	2	$2^2 + 2^2$	8
2	3	$2^3 + 3^2$	17
3	2	$3^2 + 2^3$	17
2	4	$2^4 + 4^2$	32
3	3	$3^3 + 3^3$	54
2	5	$2^5 + 5^2$	57

$…8,\ 17,\ 32,\ 54,\ 57,\ 100,\ 145,\ 177,\ 320,\ 368,\ \dots$

$x^y + y^x$ is symmetric, so swapping $x$ and $y$ gives the same number.
They grow very rapidly as $x$ and $y$ increase.
Some Leyland numbers are prime, for example:
- 17
- 100 (not prime, but notable as $4^2 + 2^4$ )
There are only a few known Leyland primes, and they are rare.