# Finding primes given a condition

by galois427
Tags: condition, primes
 P: 16 Hi, I need help solving this problem. The question asks me to find all prime numbers p such that p^2 = n^3 + 1 for some integer n.
 Sci Advisor HW Helper P: 1,123 This is really quite easy as p^2 can only be p times p or 1 times p^2. Now, $$n^3 + 1 = (n + 1)(n^2 - n + 1)$$ So one possible solution is where one of the factors is equal to 1 and the other is equal to p^2. Or when they are both equal to each other. Test those out and you should find all the possible solutions for n.
 P: 646 An alternative, all odd p^2 is usually of the form p^2 = 1 mod 4 n^3+1 = 1 mod 4 or n^3 = 0 mod 4 Now this is useful if and only if u have a list of primes ... So first u can generate a list and then check for conditions ... -- AI

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