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Continuants and some decompositions into squares
journal contribution
posted on 2015-01-01, 00:00 authored by C Delorme, Guillermo Pineda VillavicencioGuillermo Pineda VillavicencioIn 1855, H. J. S. Smith proved Fermat’s two-square theorem using the notion of
palindromic continuants. In his paper, Smith constructed a proper representation
of a prime number p as a sum of two squares, given a solution of z2+1 ⌘ 0 (mod p),
and vice versa. In this paper, we extend the use of continuants to proper representations by sums of two squares in rings of polynomials over fields of characteristic
di↵erent from 2. New deterministic algorithms for finding the corresponding proper
representations are presented.
Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic forms.
palindromic continuants. In his paper, Smith constructed a proper representation
of a prime number p as a sum of two squares, given a solution of z2+1 ⌘ 0 (mod p),
and vice versa. In this paper, we extend the use of continuants to proper representations by sums of two squares in rings of polynomials over fields of characteristic
di↵erent from 2. New deterministic algorithms for finding the corresponding proper
representations are presented.
Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic forms.
