Gematria

Gematria, or numerology, therefore fascinated the Hellenes, that in ancient Greece the numbers were represented by the letters of the alphabet and each number corresponded to a certain number. Comparing these numbers, corresponding to their names, comparing their properties, the Greeks established many curious dependencies, which continue to attract the attention of mathematicians to this day. Thus, numbers equal to the sum of their divisors are called perfect. This is: 6 = 1 + 2 + 3=2*3=2*(2²-1); 28=1 + 2 + 4 + 7 + 14 = 2²*7; 96 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 2⁴*31=2^4*31=2⁴*(2⁵-1). All the now-known perfect numbers are even. The question of the existence of odd perfect numbers has not yet been resolved.

Another problem that gematria considers is friendly numbers: the sum of the dividers of the first must equal the second and vice versa. The Greeks found only one such pair: 220 and 284. Indeed, the number of 220-11 dividers is 1, 2, 4, 5, 10, 20, 11, 22, 44, 55, 110. If you add them, you get just 284 and this second the number of 5 divisors - 1, 2, 4, 71, 142. Their sum is 220. Only in the 17th century the famous P. Pharma (1601-1665) managed to find a new pair of friendly numbers: 17296 and 18416. Now with the help of computer within million found 42 couples friendly numbers. And what happened? The numbers in these pairs were either even or odd, but there is not a single pair where one number is even and the other odd!