Discussion Overview
The discussion revolves around the existence of infinitely many primes of the form p² + nq², where both p and q are prime numbers. Participants explore the implications of this claim, particularly in relation to specific values of n and the nature of the resulting numbers.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants reference a paper claiming there are infinitely many primes of the form p² + nq², questioning whether this means all such numbers are prime.
- Others clarify that the claim does not imply all numbers of that form are prime, but rather that there are infinitely many primes among them.
- One participant suggests that the existence of infinitely many primes implies there must also be infinitely many non-primes, proposing that gaps between primes exist.
- Another participant hints at specific values of n (such as n=4 and n=6) that could lead to non-prime results, indicating that the expression can yield even numbers or numbers divisible by 3.
- A participant expresses uncertainty about their mathematical abilities and requests a proof regarding the existence of non-primes in this context.
- There is a clarification regarding the nature of n as a positive integer variable, with specific values being discussed.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the implications of the claim regarding primes and non-primes. There are competing views on whether the existence of infinitely many primes necessitates the existence of infinitely many non-primes.
Contextual Notes
Participants express varying levels of mathematical understanding, and there are unresolved questions about the proofs related to the claims made. The discussion includes specific examples and values that may influence the outcomes of the expressions discussed.