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Can a Lucas number be a sum of three repdigits?

Chèfiath A. Adegbindin, Alain Togbé (2020)

Commentationes Mathematicae Universitatis Carolinae

We give the answer to the question in the title by proving that L 18 = 5778 = 5555 + 222 + 1 is the largest Lucas number expressible as a sum of exactly three repdigits. Therefore, there are many Lucas numbers which are sums of three repdigits.

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

Roman Wituła, Damian Słota (2006)

Open Mathematics

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.

Circulants and the factorization of the Fibonacci–like numbers

Jaroslav Seibert, Pavel Trojovský (2006)

Acta Mathematica Universitatis Ostraviensis

Several authors gave various factorizations of the Fibonacci and Lucas numbers. The relations are derived with the help of connections between determinants of tridiagonal matrices and the Fibonacci and Lucas numbers using the Chebyshev polynomials. In this paper some results on factorizations of the Fibonacci–like numbers U n and their squares are given. We find the factorizations using the circulant matrices, their determinants and eigenvalues.

Common terms in binary recurrences

Erzsébet Orosz (2006)

Acta Mathematica Universitatis Ostraviensis

The purpose of this paper is to prove that the common terms of linear recurrences M ( 2 a , - 1 , 0 , b ) and N ( 2 c , - 1 , 0 , d ) have at most 2 common terms if p = 2 , and have at most three common terms if p > 2 where D and p are fixed positive integers and p is a prime, such that neither D nor D + p is perfect square, further a , b , c , d are nonzero integers satisfying the equations a 2 - D b 2 = 1 and c 2 - ( D + p ) d 2 = 1 .

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