On the relative sizes of \(A\) and \(B\) in \(p=A^2+B^2\), where \(p\) is a prime \(\equiv 1\pmod 4\) (Q2710509)

Let \(p\) be a prime \(\equiv 1\bmod 4\) such that the norm of the fundamental unit \(T+U\sqrt{2p}\) of the real quadratic field \(\mathbb{Q}(\sqrt{2p})\) is \(-1\). Let \(L\) denote the length of the period of the continued fraction expansion of \(\sqrt{2p}\). Let \(p= A^2+B^2\), \(A\equiv 1\pmod 2\), \(B\equiv 0\pmod 2\) be the unique representation of \(p\) as a sum of two positive integers. Then the following theorem is proved: NEWLINE\[NEWLINEA>B \quad\text{if and only if}\quad L\equiv T\bmod 4.NEWLINE\]NEWLINE The proof is based on an earlier paper of the author and \textit{P. Kaplan} [J. Number Theory 23, 169-182 (1986; Zbl 0596.10013)].