On the Diophantine equation (x^m-1)/(x-1) = (yⁿ-1)/(y-1)

On the diophantine equation x3-1x-1=ynYuan1 2005Journal of Number Theory7000View full textAdd to dashboardCite

“…Makowski and Schinzel [10,11] proved several special cases of this conjecture. Nagell [12] confirmed a conjecture of Ramanujan [16] that the equation (3,4), (5,5), (11,7), (181,15). This implies that the solutions of Eq.…”

“…If (x, y, m, n) satisfy (1), then max(x, y, m, n) is bounded by an effectively computable number depending only on r and s. This is the first result of the type where there is no restriction on bases x and y and the exponents m and n extend over an infinite set. Bugeaud and Shorey [4] showed that the above assertion continues to be valid when the ratio (m − 1)/(n − 1) is bounded.…”

“…[7] contains an error, as pointed out by Leu and Li [9]. Furthermore, the error in [7] led to an error in [4] as well. However, these errors do not invalidate the present paper.…”

On a problem of Diophantus for higher powers

“…considered, respectively, in [15], [13] and [4]. The numerical value we get in Theorem 1 is remarkably small.…”

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

“…The only known solutions are (x, y, m, w) = (2, 5, 5, 2) and(2, 90,13,3) and it is conjectured that there are no other solutions. In order to deal with case (B) of Corollary 41, we prove the following new result on Goormaghtigh's equation.Theorem 43.…”