ON THE SEQUENCES $\{a_n\}$ AND $\{b_n\}$ DEFINED BY $p^n=a_n^2 + b_n^2$, WHERE $p$ IS A PRIME OF THE FORM $p \equiv 1 (\text{mod} 4)$

Abstract

Let $p \equiv 1$ (mod 4) be a prime number, and consider its representation as a sum of two squares, $$p=a^2 + b^2,$$ for some integers $a$ and $b$. In this paper, we study the sequences $\{a_n\}$ and $\{b_n\}$ defined by the relation $$p^n=a_n^2 + b_n^2,$$ arising from powers of the Gaussian integer $a+bi$. In Section 2, we investigate the arithmetic structure of these sequences, proving that gcd$(a_n, b_n)=1$ for all $n$, and establishing the uniqueness of the representation $p^n=a_n^2 + b_n^2$ under coprimality conditions. In Section 3, we study the quadratic residuacity of the terms $a_n$ and $b_n$ modulo $p$, showing how it depends on the residue class of $p$ (mod 8) and describing the distribution of residues and non-residues along the sequences. Finally, in Section 4, we analyze the periodic behavior of $(a_n, b_n)$ modulo $p$, proving that the sequence of ordered pairs is periodic with period equal to the multiplicative order of 2$a$ modulo $p$, and deriving further congruence properties of $a_n$ and $b_n$.