Pythagoras: Fun facts

Numbers to the Pythagorians were like the ideal forms sought after by Plato. Everything could be explained by their magic properties.

According to Pythagoras, numbers had a real and separate existence outside our minds.

10 was a perfect number, according to Pythagoras.

http://home.c2i.net/greaker/comenius/9899/pythagoras/pythagoras.html

http://www.samos.net/pithagoras.htm

10 is a triangular number: 1 + 2 + 3 + 4 = 10
10 is the only triangular number which is also the sum of 2
1^2 + 3^2 = 10 (consecutive square odd numbers)
10 cannot be the difference of 2 squares, because 10 is of the form 4n + 2
10 = 2 + 3 + 5 (sum of the first 3 primes)
10! = 6! * 7! = 3! * 5! * 7!
(Unique solution to the factorial equation n! = a! * b! * c! with consecutive prime factors)
(10!)^2 + 1 is a prime number

Pythagorean triples

http://en.wikipedia.org/wiki/Pythagorean_triple


A little fun with Pythagorean triples:

Exactly four right triangles with integer side lengths exist with a leg equal to 15. These are:

(15, 112, 113), (15, 20, 25), (15, 36, 39), and (8, 15, 17).

How many different right triangle with integer side lengths exist with a leg equal to 42? How do you know?

BONUS QUESTION:

42 is the product of 3 different prime numbers - 2, 3, and 7.

Resolve this problem given a leg which is the product of n different primes, given that, as in this problem, one of these primes is 2.