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Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

Let ${(L_{n})}_{n \geq 0}$ be the Lucas sequence. We show that the Diophantine equation $L_{n} - L_{m} = M_{k}$ has only the nonnegative integer solutions $(n, m, k) = (2, 0, 1)$ , $(3, 1, 2)$ , $(3, 2, 1)$ , $(4, 3, 2)$ , $(5, 3, 3)$ , $(6, 2, 4)$ , $(6, 5, 3)$ where $M_{k} = 2^{k} - 1$ is the $k$ th Mersenne number and $n > m$ .

Multivariate Diophantine equations with many solutions

J.-H. Evertse, P. Moree, C. L. Stewart, R. Tijdeman (2003)

Acta Arithmetica

Currently displaying 1 – 2 of 2

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