Читать книгу The Little Book of Mathematical Principles, Theories & Things - Robert Solomon - Страница 15

A number is perfect if it equals the sum of its proper divisors.

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The search continues for perfect numbers, especially an odd perfect number.

Numerology is the magical side of mathematics and some traces – such as perfect numbers – remain in modern mathematics. Perfect numbers were thought to be mystically superior to all others and this can be seen by the following quotation from St Augustine’s City of God (420 AD):

Six is a perfect number, not because God created the world in six days, rather the other way round. God created the world in six days because six is perfect…

A perfect number is equal to the sum of its proper divisors. The first two perfect numbers are 6 and 28.

The divisors of 6 are 1, 2, and 3.

6 = 1 + 2 + 3

The divisors of 28 are 1, 2, 4, 7, and 14.

28 = 1 + 2 + 4 + 7 + 14

The next perfect numbers are 496 and 8128, the only ones known before the 13th century. The next three were found (along with three incorrect numbers) by Arab mathematician Ibn Fallus.

Finding even perfect numbers is comparatively easy. There is a formula for them, which essentially appears in Euclid’s Elements. The formula is 2^n–1(2ⁿ – 1), provided that the term inside the brackets is a prime number.

All the perfect numbers that have so far been discovered are even; an odd perfect number, if it exists, remains to be found. This is the oldest unsolved problem in mathematics.

Certainly there are no odd perfect numbers up to 10³⁰⁰ (1 followed by 300 zeros). They may not exist, but if one is ever found, mathematicians will already know a lot about it: that it has at least nine prime factors, for example.