Solving exponential diophantine equations using lattice basis reduction algorithms

B.M.M. de Weger

Research output: Contribution to journalArticleAcademic

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Abstract

Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < yδ in x, y S for fixed δ (0, 1), and for the diophantine equation x + Y = z in x, y, z S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented.
Original languageEnglish
Pages (from-to)325-367
JournalJournal of Number Theory
Volume26
Issue number3
DOIs
Publication statusPublished - 1987

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Exponential Diophantine Equations
Lattice Basis Reduction
Diophantine Inequalities
Diophantine Approximation
Diophantine equation
P-adic
Divisor
Finite Set
Integer

Cite this

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title = "Solving exponential diophantine equations using lattice basis reduction algorithms",
abstract = "Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < yδ in x, y S for fixed δ (0, 1), and for the diophantine equation x + Y = z in x, y, z S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented.",
author = "{de Weger}, B.M.M.",
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language = "English",
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journal = "Journal of Number Theory",
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Solving exponential diophantine equations using lattice basis reduction algorithms. / de Weger, B.M.M.

In: Journal of Number Theory, Vol. 26, No. 3, 1987, p. 325-367.

Research output: Contribution to journalArticleAcademic

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N2 - Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < yδ in x, y S for fixed δ (0, 1), and for the diophantine equation x + Y = z in x, y, z S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented.

AB - Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < yδ in x, y S for fixed δ (0, 1), and for the diophantine equation x + Y = z in x, y, z S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented.

U2 - 10.1016/0022-314X(87)90088-6

DO - 10.1016/0022-314X(87)90088-6

M3 - Article

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SP - 325

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JF - Journal of Number Theory

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