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...
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