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On the Diophantine
equation (xn - 1)/(x-1) = yq
Yann Bugeaud, Maurice Mignotte and Yves Roy |
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Vol. 193 (2000), No. 2, 257–268
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Abstract |
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We prove that if (x,y,n,q)≠(18,7,3,3) is a solution of the Diophantine equation (xn − 1)∕(x− 1) = yq with q prime, then there exists a prime number p such that p divides x and q divides p − 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy computer calculations. |
Milestones
Received: 25 September 1998
Published: 1 April 2000
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Authors |
| Yann Bugeaud | |
| Université Louis Pasteur 67084 Strasbourg France |
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| Maurice Mignotte | |
| Université Louis Pasteur 67084 Strasbourg France |
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| Yves Roy | |
| Université Louis Pasteur 67084 Strasbourg France |
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