Continuants and some decompositions into squares

Details Continuants and some decompositions into squares Continuants and some decompositions into squares

2011-12-20

arxiv.org/abs/1112.4535v1

Mathematics

Number Theory

16 pages

Scientific paper

In 1855 H. J. S. Smith proved Fermat's Two Squares using the notion of palindromic continuants. In his paper Smith constructed a proper representation as a sum of two squares of a prime number $p$, given a solution of $z^2+1\equiv0\pmod{p}$, and vice versa. In this paper we extend Smith's approach to proper representations by sums of two squares in rings of polynomials on fields of characteristic different from 2. Our approach will also work for other representations of integers, such as sums of four squares. We keep as far as possible the palindromic character of the representations. While our results are likely not new, we believe our extension of Smith's approach is new.

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