### Bo He

Sichuan Provincial Key Laboratory of Computer Software, Sichuan Normal University, Chengdu, 610068, P. R. China
e-mail: hebo-one@hotmail.com
e-mail: bhe@live.cn

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,

has only the solutions (x,y,n) = (5,2,5) and (90,2,13).

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

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

Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.01

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