The Number Theory:
- Pierre de Fermat spent a great deal of his time studying the properties of the aliquot parts, or proper divisors, of the natural numbers. It was known to the Greeks that 220 and 284 were a pair of “friendly,” or amicable, numbers. In modern notation, using the number theoretic function sigma, we write are amicable numbers. This method produces p, q, and r prime when n = 1, 2, 4, and 7. Fermat surely knew that when n = 1 the two numbers (34 and 20) are not amicable. The following table shows the amicable numbers generated for n = 2, 4, and 7. Fermat is generally credited with the case n = 4, while Descartes is credited with the case n = 7. These are the only three pairs of amicable numbers that can be found by this algorithm for n < 20000
- The Fermat number, named after Pierre Format is a positive integer of the form Fn=2^2n+1
where n is a non negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 im OEIS). If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. (If n = ab where 1 ≤ a, b ≤ n and b is odd, then 2n + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0, F1, F2, F3, and F4.