There seems to be no shortage of people finding patterns related to prime numbers on this forum. Alas, I felt the need to share my stumbling upon such an instance.
Take a list of primitive Pythagorean triples, let their sides be denoted by the common a , b , and c .
I have noticed that the difference ( a-b with a>b) between a and b is very, very often a prime number, which holds for large values. And, if the number is not prime, then it is semi-prime (only factors are 1, itself, and two primes).
3/4/5= 1 .... 1428/1475/2053= 47 .... 693/1924/2045= 1231 20/21/29=1 .... 1204/1653/2045= 449 .... 1281/1640/2081= 349 5/12/13= 7 8/15/17= 7 7/24/25= 17 12/35/37= 23 9/40/41= 31
Of course, there are those that fail (are semi-prime; I don't believe I've found any counterexamples).
For example: 819/1900/2069= 1081 (23*47).
I find it intriguing that it outputs so many primes. I am not sure which fields of mathematics this relates to, but I would love to have the input of others. Is this only an interesting phenomenon or a gold nugget?
Pythagorean Triples up to 2100 + link up to 10000: http://www.tsm-resources.com/alists/trip.html
My conjecture: : Every number generated in this way is a prime or semi-prime number. And: note that this only includes primitive triples.