Results 1 to 10 of about 202 (150)
A New Solution to a Cubic Diophantine Equation
Axioms, 2022 A positive integer, which can be written as the sum of two positive cubes in two different ways, is known as a “Ramanujan number”. The most famous example is 1729=103+93=123+13, which was identified by Ramanujan as the lowest such number.
John C. Butcher
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A system of cubic diophantine equations [PDF]
Journal of Number Theory, 1977 AbstractIn this paper we solve x3 + y + 1 − xyz = 0 completely and study a pair of simultaneous cubic diophantine equations (1) x | y3 + 1 and y | x3 + 1, where x and y are positive integers. The main result in this paper is that there exist an infinite number of sequences such that x and y satisfy (1) if and only if they are consecutive terms of one ...
Mohanty, S.P.
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Solving elliptic diophantine equations: the general cubic case [PDF]
Acta Arithmetica, 1999 The Elliptic Logarithm Method for computing explicitly all integral solutions of a diophantine equation that defines an elliptic curve over \(\mathbb Q\) (or, more generally, over a number field) as a practical method, has been developed by \textit{R. J. Stroeker} and \textit{N. Tzanakis} [Acta Arith.
Stroeker, Roelof J. +1 more
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Diophantine equations arising from cubic number fields [PDF]
Journal of Number Theory, 1981 AbstractThe solutions to a certain system of Diophantine equations and congruences determine, and are determined by, units in galois cubic number fields. These solutions fall into two classes: certain ones determine infinite families of solutions, while others do not. We construct an infinite number of examples of each type of solution. We obtain these
Thomas, E, Vasquez, A.T
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The Solubility of Diagonal Cubic Diophantine Equations [PDF]
Proceedings of the London Mathematical Society, 1999 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Heath-Brown, David, Heath-Brown, DR
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Optimization of the multivariate polynomial public key for quantum safe digital signature [PDF]
Scientific Reports, 2023 Kuang, Perepechaenko, and Barbeau recently proposed a novel quantum-safe digital signature algorithm called Multivariate Polynomial Public Key or MPPK/DS.
Randy Kuang, Maria Perepechaenko
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The integer solutions of the cubic Diophantine equation x3±33=pqy2
Xi'an Gongcheng Daxue xuebao, 2021 The solvability of a class of cubic Diophantine equations is studied by using properties of congruence, Legendre symbol and the methods of elementary number theory.
Heng LI, Hai YANG, Yongliang LUO
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Rocky Mountain Journal of Mathematics, 2002 A necessary and sufficient condition for the solvability of the simultaneous Diophantine equations \(C_1(x,y) = C_1(u,v)\) and \(C_2(x,y) = C_2(u,v)\) is provided, where \(C_i(x,y)\), \(i = 1, 2\) are arbitrary binary cubic forms. If the forms \(C_i(x,y)\) have a common factor, the complete solution of these equations is obtained; otherwise, infinitely
Ajai Choudhry
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Complete solutions to a family of cubic Diophantine equations
Journal of Number Theory, 1990 It is proved in this paper that if \(n\geq 1.365\cdot 10^7\), then the Diophantine equation \[ f_n(x,y)=x^3-(n-1)x^2y-(n+2)xy^2-y^3=\pm 1, \tag{1} \] has only the ``trivial'' solutions \((\pm 1,0)\); \((0,\pm 1)\); \((\pm 1,\mp 1)\). The proof follows that standard method, which is used for the derivation of an effective upper bound for the solutions ...
Emery Thomas
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On some classes of homogeneous ternary cubic diophantine equations
Arkiv for Matematik, 1975 A homogeneous ternary cubic equation can in general be taken by a real transformation into the canonical form \[ ax^3+ by^3 + cz^3 = dxyz.\tag{1} \] The author considers the possibility of doing so by a rational transformation; and for certain classes of equations, involving two or more parameters, he gives simple necessary and sufficient conditions ...
Erik Dofs
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