Glasnik Matematicki, Vol. 44, No.1 (2009), 1-6.

A REMARK ON THE DIOPHANTINE EQUATION (x³-1)/(x-1)=(yⁿ-1)/(y-1)

Bo He

Abstract.   In this remark, we use some properties of simple continued fractions of quadratic irrational numbers to prove that the equation

(x³ - 1)/(x - 1) = (yⁿ - 1)/(y - 1),   x, y, n N, x > 1, y > 1, n > 3, n odd,

2000 Mathematics Subject Classification.   11A55, 11D09, 11D61.

Key words and phrases.   Goormaghtigh equation, continued fraction, congruence.

DOI: 10.3336/gm.44.1.01

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