# Finding primes given a condition

1. Oct 8, 2004

### galois427

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.

2. Oct 9, 2004

### Zurtex

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.

3. Oct 13, 2004

### TenaliRaman

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