We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation $- 1^{k} + 2^{k} - \dots + {(- 1)}^{x} x^{k} = g (y)$ with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.

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On some generalizations of the diophantine equation s ( 1 k + 2 k + ⋯ + x k ) + r = d y n

On the diophantine equation x(x-1)...(x-(m-1)) = λy(y-1 )...(y-(n-1)) + l

Diophantine equations with Euler polynomials

On some generalizations of the diophantine equation $s (1^{k} + 2^{k} + \dots + x^{k}) + r = d y^{n}$