On simultaneous representations of primes by binary quadratic forms

Joseph B. Muskat

doi:10.1016/0022-314x(84)90111-2

On simultaneous representations of primes by binary quadratic forms

Joseph B. Muskat

Department of Computer Science and Artificial Intelligence

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to "predict" whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

Original language	English
Pages (from-to)	263-282
Number of pages	20
Journal	Journal of Number Theory
Volume	19
Issue number	2
DOIs	https://doi.org/10.1016/0022-314x(84)90111-2
State	Published - Oct 1984

Bibliographical note

Funding Information:
* This research received some support from National

Funding

* This research received some support from National

Funders	Funder number
Russian Science Foundation	GP8973

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10.1016/0022-314x(84)90111-2

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abstract = "Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to {"}predict{"} whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.",

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AB - Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to "predict" whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

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On simultaneous representations of primes by binary quadratic forms

Abstract

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