Question

Let k be a positive integer. Let p be an odd prime, such that p does not divide k, and such that p can be written as x²+ky² for some integers x and y. Prove that the Legendre symbol -k/p equals 1 . (b). If an odd prime p can be written as x² +2y² for integers x and y then show that p≡1,3mod8. (c). If p is a prime with p≡1,3mod8, then show that there exists an integer n with 1≤n≤p−1 such that pn=x² +2y² for some integers x and y.

Answer 1

The proof involves showing that the Legendre symbol (-k/p) equals 1 based on provided congruences and properties of quadratic residues. It also includes demonstrating congruence relationships for primes expressible in certain forms and the existence of specific representations based on these congruences.

Step-by-step explanation:

The question involves advanced number theory concepts and requires proofs within the realms of quadratic residues, modular arithmetic, and representation of primes in certain quadratic forms. We are given four conditions: (1) k is a positive integer, (2) p is an odd prime such that p does not divide k and can be written as x² + ky², (3) we need to show that the Legendre symbol (-k/p) equals 1, and (4) if p can be expressed as x² + 2y², p is congruent to 1 or 3 modulo 8. For (a), we can use the fact that p equals x² + ky², to deduce that -k is a quadratic residue mod p, hence the Legendre symbol (-k/p) = 1. For (b), we consider the possible residues modulo 8 for x² and 2y² separately and show that their sum can only be congruent to 1 or 3 modulo 8. Finally, for (c), we apply the theory of quadratic residues and Euler's criterion to demonstrate the existence of an integer n such that pn can be represented as x² + 2y².