On A Theorem of Niven

Williams, Kenneth S.

doi:10.4153/cmb-1967-055-3

Cited by 3 publications

(2 citation statements)

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“…h = 1) appeared in [28,34,52]. The number of representations of non-zero Gaussian integers as sums of two Gausssian integers was obtained by Pall [36], and later by Williams [48,50]. Elia [15] proved that a totally positive integer…”

Section: Introductionmentioning

confidence: 99%

Continuants and some decompositions into squares

Delorme,

Pineda-Villavicencio

2011

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In 1855 H. J. S. Smith [8] 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 z 2 +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 different from 2. New deterministic algorithms for finding the corresponding proper representations are presented.Our approach will provide a new constructive proof of the foursquare theorem and new proofs for other representations of integers by quaternary quadratic forms.

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Section: Introductionmentioning

confidence: 99%

Continuants and some decompositions into squares

Delorme,

Pineda-Villavicencio

2011

Preprint

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“…This result was rediscovered (using a different method) by the author [3]. Using ideas from [2], [3] we give a very simple proof of Pall's theorem.…”

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confidence: 95%

Note on a theorem of Pall

Williams¹

1971

Proc. Amer. Math. Soc.

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Diophantine equationsx2-Dy2=-1,±2, odd graphs, and their applications

2005

Journal of Number Theory

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Let D > 2 be a square-free integer and define a direct graph G(D) such that the vertices of the graph are the primes p i dividing D, and the arcs are determined by conditions on the quadratic residues (p i /p j ). In this paper, our main result is that x 2 − Dy 2 = k, where k = −1, ±2, is solvable if the corresponding graph is "odd". Being "odd" is a complicated technical condition but we obtain a new criterion for the solvability of these diophantine equations which is quite different from that obtained by Yokoi in 1994. The solvability of these diophantine equations are related (by a theorem of Moser) to the stufe (the minimal number of squares −1 is the sum of integral squares) of an imaginary quadratic number field. We obtain an explicit result that the stufe is 2. Finally, we easily prove some results (originally proved by Hsia and Estes) on the expressibility of integers in an imaginary quadratic number field as sums of 3 integral squares.

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On A Theorem of Niven

Cited by 3 publications

References 1 publication

Continuants and some decompositions into squares

Continuants and some decompositions into squares

Note on a theorem of Pall

Diophantine equationsx2-Dy2=-1,±2, odd graphs, and their applications

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