Finding primes given a condition

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SUMMARY

The discussion focuses on finding prime numbers \( p \) such that \( p^2 = n^3 + 1 \) for some integer \( n \). The factorization \( n^3 + 1 = (n + 1)(n^2 - n + 1) \) is crucial for identifying potential solutions. The discussion highlights that one factor must equal 1 while the other equals \( p^2 \) or both factors can be equal. Additionally, it notes that all odd \( p^2 \) is typically of the form \( p^2 \equiv 1 \mod 4 \), which leads to the condition \( n^3 + 1 \equiv 1 \mod 4 \).

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  • Basic knowledge of algebraic factorization
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  • Learn about modular arithmetic and its applications in number theory
  • Study algebraic identities and their factorization techniques
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galois427
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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.
 
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This is really quite easy as p^2 can only be p times p or 1 times p^2.

Now, [tex]n^3 + 1 = (n + 1)(n^2 - n + 1)[/tex]

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.
 
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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