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The equation xn−1/x−1=yq with x square

Published online by Cambridge University Press:  01 January 1999

N. SARADHA  and
T. N. SHOREY
N. SARADHA
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
T. N. SHOREY
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India
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Abstract

It has been conjectured that equation

formula here

has only finitely many solutions. We observe that (x, y, n, q)=(3, 11, 5, 2), (7, 20, 4, 2) and (18, 7, 3, 3) are solutions of (1). These are the only known solutions and perhaps (1) has no other solution. Ljunggren [8] proved in 1943 that (1) with q=2 has no solution other than x=3, y=11, n=5 and x=7, y=20, n=4. Shorey and Tijdeman [16] confirmed the conjecture if x is fixed. Let z>1 be an integer. The main purpose of this paper is to show that (1) has no solution if x is restricted to the infinite set of squares z2 with z[ges ]32 and z∈{2, 3, 4, 8, 9, 16, 27}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 125 , Issue 1 , January 1999 , pp. 1 - 19
Copyright
Cambridge Philosophical Society 1999

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