<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study the Diophantine equation <italic>x</italic>2 = <italic>n</italic>2 + <italic>mn</italic> + <italic>np</italic> + 2<italic>mp</italic> with <italic>m</italic>, <italic>n</italic>, <italic>p</italic>, and <italic>x</italic> being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: <italic>a</italic>2 + <italic>b</italic>2, <italic>a</italic>2 + 2<italic>b</italic>2, and 2<italic>a</italic>2 − <italic>b</italic>2. We show that there are infinitely many solutions and conjecture that there are always solutions if <italic>x</italic> ≥ 5 and <italic>x</italic> ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.</abstract>

In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.

Intersecting semi-disks and the synergy of three quadratic forms

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Intersecting semi-disks and the synergy of three quadratic forms"}

In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry...

Intersecting semi-disks and the synergy of three quadratic forms

Published Online: Sep 26, 2019

Page range: 5 - 14

Received: Jul 28, 2018

Accepted: Dec 18, 2018

DOI: https://doi.org/10.2478/auom-2019-0016

Keywords
Diophantine equation, Prime numbers, Quadratic forms

© 2019 Andrew D. Ionaşcu, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Intersecting semi-disks and the synergy of three quadratic forms

Published Online: Sep 26, 2019

Page range: 5 - 14

Received: Jul 28, 2018

Accepted: Dec 18, 2018

DOI: https://doi.org/10.2478/auom-2019-0016

KeywordsDiophantine equation, Prime numbers, Quadratic forms

© 2019 Andrew D. Ionaşcu, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Diophantine equation, Prime numbers, Quadratic forms