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June 3rd, 2014, 03:18 PM  #1 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Congruences and diophantine equations
Hi everybody, Let n positive integer > 1 p and q odd distinct prime numbers Find n, p and q such as : (p^n mod pq) + (q^n mod pq) = p + q I know that there are an infinite number of solutions. Is there some theorems to build? Is there a way to factorize some odd semiprime numbers? 
June 6th, 2014, 04:00 PM  #2 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
Hi, No answer yet Is there something wrong? 
June 6th, 2014, 06:13 PM  #3 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53 
Well, no, you can't factor semiprimes quickly, otherwise the RSA algorithm would all break down, and cryptography would be utterly ruined. As for the original equation, I'm not sure whether it can be simplified. 
June 7th, 2014, 05:40 AM  #4  
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Quote:
So why are presuming so?  
June 7th, 2014, 07:24 AM  #5 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
Here you can read in french the article about the discrete logarithm Actualité > En bref : un nouvel algorithme déjoue les systèmes de cryptographie or here in english : A Heuristic QuasiPolynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic  Springer The breach discovery is very recent. 

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