Magical numbers

This is no talisman, but there is indeed magic in numbers.

Friendly numbers (Amicable numbers)

These are pairs of numbers such that each number is the sum of the factors of the other number (including 1, but excluding the number itself). The pair (220, 284) was discovered by the Pythagoreans. In 1636, Fermat discovered (17296, 18416). Descartes located a third pair (9363584, 9437056). Euler made a list of 62 friendly pairs. Curiously, a much smaller pair (1184, 1210) was overlooked by all these giants and was discovered by a sixteen-year-old Italian, Nicolo Paganini.

Sociable numbers

These are sets of three or more numbers which form a closed loop, that is, the factors of the first number add up to the second, factors of the second add up to the third and so on, until the divisors of the last add up to the first. (12496, 14288, 15472, 14536, 14264) is a set of sociable numbers.

Euler conjecture

Similar to the Fermat’s last theorem which stated that

xⁿ + yⁿ = zⁿ

where n, x, y and z are whole numbers is true, only for n not greater than 2, Euler also conjectured that the equation

x⁴ + y⁴ + z⁴ = w⁴

doesn’t have any solutions. For two hundred years, nobody could prove the conjecture, but on the other hand, nobody could disprove it either. This was finally cracked in 1988, when Naom Elkies discovered the following solution,

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴

^{Source: Fermat's Last Theorem by Simon Singh}