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Cubic and Quartic Diophantine Equations [PDF]

, 2020
In Chapters 3 and 4 we were concerned with quadratic equations in two variables, but were only allowing ourselves integer solutions. An equation involving polynomials together with the constraint that we are only interested in integer solutions is called a Diophantine equation. In this sense we have been considering ‘quadratic Diophantine equations’.
Menny Aka, Manfred Einsiedler, Thomas Ward   +2 more
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Diophantine Equations and the Cubic Ambiguity Resolution Algorithm. [PDF]

, 1982
Abstract : The ambiguity resolution problem for the TOAME four phase interferometer system is presented in terms of the integer solutions of systems of linear equations. (Author)
null Jr, James T. Cater
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On Some Cubic Diophantine Equations

American Journal of Mathematics, 1943
exaly   +3 more sources

Cubic Diophantine Equations with Reducible Cubic Part

Proceedings of the London Mathematical Society, 1970
Summary: An equation of the type considered can, by suitably transforming the variables, written as \[ yQ + Q_1 + L + N = 0, \tag{1} \] where \(N\) is an integer and \(Q,Q_1, L\) are forms of degrees \(2,2,1\) in \(n\) variables \(x_1,\ldots,x_{n-1}, y\).
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Cubic Diophantine equations: the necessary congruence condition

Mathematika, 1965
1. Let p be a prime, t a positive integer, and ϕ a cubic polynomial with integral coefficients, and an integral constant term.
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Cubic Diophantine equations: a supplementary congruence condition

Mathematika, 1965
For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3),
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A PORTRAYAL OF INTEGER SOLUTIONS TO NON-HOMOGENEOUS TERNARY CUBIC DIOPHANTINE EQUATION

2023
This paper is concerned with the problem of determining varieties of non-zero distinct integer solutions to the non-homogeneous ternary cubic diophantine equation; Different sets of integer solutions to the above equation are obtained by reducing it to the equation, which is solvable, through employing suitable transformations.
J. Shanthi, M. A. Gopalan
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Small Prime Solutions to Cubic Diophantine Equations

Canadian Mathematical Bulletin, 2013
Abstract.Let a1,;… a9 be nonzero integers and n any integer. Suppose that a1+…+a9 ≡ n (mod 2) and (ai ; aj ) = 1 for 1 ≤ i < j ≤9. In this paper we prove the following:(i) If aj are not all of the same sign, then the cubic equation has prime solutions satisfying pj ≪|n|1/3 + max{|aj|}14+∊.(ii) If all aj are positive and n ≫ max{|aj|} 43+∊, then is
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