Author
Paul Voutier
Other affiliations: City University London
Bio: Paul Voutier is an academic researcher from University of Colorado Boulder. The author has contributed to research in topics: Algebraic number & Divisor. The author has an hindex of 9, co-authored 31 publications receiving 769 citations. Previous affiliations of Paul Voutier include City University London.
Papers
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TL;DR: In this paper, it was shown that every Lucas and Lehmer number without a primitive divisor has a primitive primitive for any value of n ≥ 30, where n is the number of nodes in the number.
Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.
352 citations
TL;DR: The first progress was due to Schinzel and Zassenhaus as mentioned in this paper, who proved that α > 1+4−s−2 where 2s is the number of complex conjugates of α.
Abstract: In 1933, D. H. Lehmer [9] asked whether it is true that for every positive e there exists an algebraic number α for which 1 1+ c0, if α is not a root of unity; in this form the question is known as Lehmer’s problem. The first progress was due to Schinzel and Zassenhaus [16], who proved in 1965 that α > 1+4−s−2 where 2s is the number of complex conjugates of α. This impliesM(α) > 1+c1/2 for a positive absolute constant c1. In 1971, Blanksby and Montgomery [2] used Fourier analysis to make a considerable refinement upon this first result. They proved that M(α) > 1+1/(52d log(6d)). In 1978, C. L. Stewart [18] introduced a method from transcendental number theory to prove that M(α) > 1+1/(10d log d). While this result is a little weaker than the one due to Blanksby and Montgomery, the method used has since become quite important as it has produced the best results yet known, results which are significantly better than those previously known and bring us quite close to the conjectured lower bound.
131 citations
TL;DR: In this article, the problem of determining all Lucas and Lehmer sequences whose nth element does not have a primitive divisor was reduced to solving certain Thue equations, using the method of Tzanakis and de Weger.
Abstract: Stewart reduced the problem of determining all Lucas and Lehmer sequences whose nth element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for n ≤ 30. Further computations lead us to conjecture that, for n > 30, the nth element of such sequences always has a primitive divisor
99 citations
TL;DR: The authors used the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, each of which is parametrized by an integral parameter.
Abstract: We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.
66 citations
35 citations
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TL;DR: In this paper, it was shown that every Lucas and Lehmer number without a primitive divisor has a primitive primitive for any value of n ≥ 30, where n is the number of nodes in the number.
Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.
352 citations
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TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem.
Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.
263 citations
TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's last theorem.
Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.
249 citations
TL;DR: In this article, the authors developed techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms.
Abstract: In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subse- quently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.
200 citations
TL;DR: In this article, all positive integer solutions (x, y, a, b, n) of the problem of finding a positive integer solution to the problem where x 2 + 2 a ⋅ 3 b = y n with n ≥ 3 and coprime x and y.
Abstract: We find all positive integer solutions ( x , y , a , b , n ) of
x 2 + 2 a ⋅ 3 b = y n with n ≥ 3 and coprime x and y .
162 citations