Perfect powers with all equal digits but one.
Primitive divisors of certain elliptic divisibility sequences
Products of integers in short intervals
Pure powers and power classes in recurrence sequences
Quelques applications de nouveaux résultats de van der Poorten
Rationals Not Expressible as a Sum of Three Unit Fractions.
Repdigits in the base as sums of four balancing numbers
The sequence of balancing numbers is defined by the recurrence relation for with initial conditions and is called the th balancing number. In this paper, we find all repdigits in the base which are sums of four balancing numbers. As a result of our theorem,...
Representations of integers by binary forms and the rank of the Mordell-Weil group.
Singular differences of powers of -matrices
Solution complète en nombres entiers de l’équation
Solutions to xyz = 1 and x+y+z = k in algebraic integers of small degree, I
Let k ∈ ℤ be such that , where . We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.
Solving an exponential Diophantine equation
Some extensions and refinements of a theorem of Sylvester
Some problems involving powers of integers
Squares in arithmetic progression with at most two terms omitted
Squares in products from a block of consecutive integers
Squares in products in arithmetic progression with at most one term omitted and common difference a prime power
Squares in products with terms in an arithmetic progression
S-unit equations over number fields.
Superelliptic equations arising from sums of consecutive powers
Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The chief innovation...