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Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

Let ( L n ) n 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 .

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

We consider the Tribonacci sequence T : = T n n 0 given by T₀ = 0, T₁ = T₂ = 1 and T n + 3 = T n + 2 + T n + 1 + T n for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

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