Prime Numbers of the Form x^2+ny^2 and Thue's Lemma

Overview

The representation of prime numbers as quadratic forms is one of the central topics of number theory, dating back to Fermat and Euler. This study investigates prime numbers expressed as x^2+ny^2 for fixed natural number n and determines the necessary conditions for such representations. Using inequality, quadratic reciprocity, modular arithmetic, basic lemmas and Thue's lemma, we find a new methodology based on applying Thue's lemma to prove that x^2+ny^2 is divisible by a given prime number and then using again Thue's lemma and arithmetic operations we find an upper bound for x^2+ny^2 that is at most n*p. We derive new criteria on p that determine whether every such prime number gives such a remainder, and, using divisibility, modes, and inequalities, check whether this is the only solution for a given n. This methodology shows that for a given n, there are only n possible solutions, and using mod, quadratic reciprocity, and arithmetic operations, we prove that among the remaining choices, the only suitable answer for a prime number is x^2+ny^2=p. This methodology simplifies the solution of the problem for a fixed value of n, and examples have been shown on classical results such as Fermat's two-square theorem, squaring, and two-square multiplication, as well as other cases. These results can make meaningful contributions to education number theory.

Competition history

• ISEF 2025 Mathematics · Entry MATH033

Resources

• ISEF abstract page ↗

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