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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.

Catalan’s conjecture

Yuri F. Bilu (2002/2003)

Séminaire Bourbaki

The subject of the talk is the recent work of Mihăilescu, who proved that the equation x p - y q = 1 has no solutions in non-zero integers x , y and odd primes p , q . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to x u - y v = 1 in integers x , y > 0 and u , v > 1 is 3 2 - 2 3 = 1 . Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

Characterizations of groups generated by Kronecker sets

András Biró (2007)

Journal de Théorie des Nombres de Bordeaux

In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus T = R / Z by subsets of Z . Here we consider new types of subgroups: let K ⊆ T be a Kronecker set (a compact set on which every continuous function f : K → T can be uniformly approximated by characters of T ), and G the group generated by K . We prove (Theorem 1) that G can be characterized by a subset of Z 2 (instead of a subset of Z ). If K is finite, Theorem 1 implies our earlier result...

Comments on the fractional parts of Pisot numbers

Toufik Zaïmi, Mounia Selatnia, Hanifa Zekraoui (2015)

Archivum Mathematicum

Let L ( θ , λ ) be the set of limit points of the fractional parts { λ θ n } , n = 0 , 1 , 2 , ⋯ , where θ is a Pisot number and λ ∈ ℚ ( θ ) . Using a description of L ( θ , λ ) , due to Dubickas, we show that there is a sequence ( λ n ) n ≥ 0 of elements of ℚ ( θ ) such that Card ( L ( θ , λ n ) ) < Card ( L ( θ , λ n + 1 ) ) , ∀ n ≥ 0 . Also, we prove that the...

Commutative algebraic groups and p-adic linear forms

Clemens Fuchs, Duc Hiep Pham (2015)

Acta Arithmetica

Let G be a commutative algebraic group defined over a number field K that is disjoint over K from a and satisfies the condition of semistability. Consider a linear form l on the Lie algebra of G with algebraic coefficients and an algebraic point u in a p-adic neighbourhood of the origin with the condition that l does not vanish at u. We give a lower bound for the p-adic absolute value of l(u) which depends up to an effectively computable constant only on the height of the linear form, the height...

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