Does a Bound L Exist for N_p x^2 - y^2?

This is not a definitive solution, and I don’t have a concrete answer. My belief is that no such bound L can exist for the equation N_p x^2 - y^2. In thinking through this, I essentially treat Quora as a forum, an invitation for a dialogue with anyone interested in exploring the problem. As a relative newcomer to Quora, my hope is that this doesn’t violate the established etiquette here.

To explore the possibility of a bound L, let’s suppose that such a bound L does exist. This would imply that for any prime p, the number N_p can be expressed as x^2 - y^2, where x and y are positive integers and y L.

Firstly, neither x nor y can be divisible by any odd prime q equal to p. If either x or y were divisible by such a prime q, both would also be divisible by q^2. This would then divide N_p, which is a square-free number, thus contradicting the assumption. Therefore, x and y can only be divisible by the prime 2 or the prime p.

Furthermore, for sufficiently large p, the smallest possible value of y would have to be a power of 2. This already seems unlikely; for very large primes p, we should not always have N_p x^2 - y^2 with y equal to a power of 2.

Assuming a bound L still exists, there are additional consequences that arise. Non-square numbers N_p must lie between two consecutive squares, say a^2 and b^2. It is known that solutions to the equation do exist for certain values. However, for sufficiently large p, we would always have a solution with x b. If x b_1, then x^2 b_1^2 b^2 - 2b_1 1. This would imply that y^2 x^2 - N_p b^2 - N_p - 2b_1 1. For sufficiently large p, this would contradict the condition that y L.

Therefore, if L exists, the difference between N_p and the closest square above it would always itself be a square for sufficiently large p. At first glance, these consequences seem unusual, yet they do not necessarily disprove the existence of L.

Although the conjectures about the existence of such a L seem remarkable, in mathematical terms, they are not proofs. The lack of definitive proof means that the question remains open, inviting further exploration and discussion within the community of number theorists or those with interest in these problems.

Given the exploration and potential implications, it is evident that Quora can be a valuable platform for initiating and participating in such mathematical discussions. The community here often provides insights and new perspectives, contributing to the advancement of mathematical knowledge.