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 January 11th, 2018, 11:08 PM #1 Member   Joined: Jan 2015 From: italy Posts: 39 Thanks: 1 Primality Test and Factorization with Pythagorean triples and quadratic diophantine A Pythagorean triple (A,b,c) with a minor cateto A odd always admits a solution (A,b,b+1) where 2*b+1=A^2 For example, let N be a semiprimo N=p*q then N/1, N/N, N/p, N/q will be our four paths Suppose we follow the N/q road then (N/q+1)/2 or (N/q-1)/2 will be odd then [(N/q+1)/2]^2+b^2=(b+1)^2 will admit integer solutions (quadratic diophantine parabolic case) or [(N/q-1)/2]^2++b^2=(b+1)^2 will admit integer solutions (quadratic diophantine parabolic case) to keep in mind that the roads that can be traveled as mentioned above are four N/1, N/N, N/p, N/q so there will be four right paths reiterating we will arrive at b=0 and find the solution q example N=67586227 solve ((((((((((((((67586227/q+1)/2-1)/2-1)/2-1)/2-1)/2)-1)/2+1)/2+1)/2-1)/2+1)/2-1)/2-1)/2)^2+b^2=(b+1)^2 , b=0 the quadratic diophantine parabolic case are obtained thus solve ((67586227/q+1)/2)^2+b^2=(b+1)^2 , b b=(4567898080095529 + 135172454*q - 3*q^2)/(8*q^2) this number 4567898080095529=N^2 this number 135172454/N= integer =2 as N=p*q then 8*b=(N^2)/(q^2)+(2*N)/(q)-3 then 8*b=p^2+2*p-3 if it admits solutions we are on a right path What do you think of this? January 12th, 2018, 09:10 AM   #2
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Quote:
 Originally Posted by gerva then N/1, N/N, N/p, N/q will be our four paths
thanks to Lutz Donnerhacke
I make this change

So we must discard example
N = 187
for N / 1 (-1, + 1, -1, -1, -1, + 1, -1)
for -N / 1 (+ 1, -1, + 1, + 1, + 1, -1, + 1)
for N / N (+ 1, + 1, + 1, + 1, + 1, + 1, + 1)
for -N / N (-1, -1, -1, -1, -1, -1, -1)

As soon as a path is different from this one must follow only that Tags diophantine, factorization, primality, pythagorean, quadratic, test, triples Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post gerva Math 23 April 27th, 2015 08:44 AM jiasyuen Number Theory 3 April 2nd, 2015 12:42 AM M_B_S Number Theory 2 July 1st, 2014 11:39 PM Pell's fish Number Theory 3 August 5th, 2011 10:38 AM julian21 Number Theory 3 November 12th, 2010 11:58 AM

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