As you may know practically all security on the internet relies on the fact that it is hard to factorise composite numbers made up of large primes.
Recently in some of my research regarding r-simplex's and prime numbers I discovered that a number N is a prime if TN(TN(N,N),5)/N is equal to an integer as far as I have tested this is not true for any composite number.
Although this is probably the slowest method known to man for finding if a number is a prime or not because TN(N,N) = pochhammer(N,N)/N! which takes a lot of ever more computing power to calculate the bigger the Value of N it does however demonstrate a direct relationship between Pn and the triangle numbers. Armed with knowledge of this special relationship something I had suspected for a long time but had never proven which I still have not actually developed a proof for but simply have tested I decided to search for a quick method. I later discvovered that using a modulation of the polygonal numbers I could generate a good chance of making a Prime. My modulation of the polygonal numbers works like this for a polygonal number with 3 sides you get the pattern {1,2,1,2,3,1,2,3,4.....} and for a polygonal number with 4 sides you get the pattern {1,2,3,1,2,3,4,5,1,2,3,4,5,6,7....} and so on. This showed me there was a link between the polygonal numbers and the prime numbers. Then with my research regarding R-Simplex's I decided to try and make use of the Jacobi symbol which I had found people used in many probable prime tests I found that the Jacobi symbol works fast with very large numbers so it ideal. I then ran into the problem that the Jacobi symbol only produces a 1 a 0 or a -1 . In order to capitalise on this I decided to modulate my polygonal modulation and came up with this. an example of this new function is Polyxmod(n,3,2) (3 being the number of sides 2 being the modulation depth) anyway this gives you the pattern {1,2,1,2,3,1,2,1,2,3,1,2,1,2,3....}. Altogether I have now found a way to deduce at least most oif the primes maybe ending up with a few composites using a function combining all of the functions.
guess(n) = {Denominator(TN(n,4)/(Polyxmod(n,5,4)-jacobisymbol(n,4)+1), Polyxmod(n,5,4)-jacobisymbol(n,5)}
then by doing some tests on this function I get these useful tests.
tests(n) = {guess(2^1*n^2),guess(2^3*n^2),guess(2^5*n^2)},{guess(2^1*n^3),guess(2^3,n^3),guess(2^5*n^3)},{guess(5^1*n^2),guess(5^3*n^2),guess(5^5*n^2)},{guess(5^1*n^3),guess(5^3*n^3),guess(5^5*n^3)},{guess(11*n),guess(11^2*n),guess(11^3*n)}. .
I know my analysis is not perfect but its getting me closer than I've ever been and if I can get it to correctly question if a number is a prime or not I think I know how to go about working out how to do factorisation by counting the occourences of integers in polyxmod(n5,5)