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
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.
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.
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