[HTML][HTML] On the Diophantine equation (x+ 1) k+(x+ 2) k+...+(2x) k= yn
In this work, we give upper bounds for n on the title equation. Our results depend on
assertions describing the precise exponents of 2 and 3 appearing in the prime factorization
of T k (x)=(x+ 1) k+(x+ 2) k+...+(2 x) k. Further, on combining Baker's method with the explicit
solution of polynomial exponential congruences (see eg [6]), we show that for 2≤ x≤ 13,
k≥ 1, y≥ 2 and n≥ 3 the title equation has no solutions.
assertions describing the precise exponents of 2 and 3 appearing in the prime factorization
of T k (x)=(x+ 1) k+(x+ 2) k+...+(2 x) k. Further, on combining Baker's method with the explicit
solution of polynomial exponential congruences (see eg [6]), we show that for 2≤ x≤ 13,
k≥ 1, y≥ 2 and n≥ 3 the title equation has no solutions.
In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T k (x)=(x+ 1) k+(x+ 2) k+...+(2 x) k. Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see eg [6]), we show that for 2≤ x≤ 13, k≥ 1, y≥ 2 and n≥ 3 the title equation has no solutions.
Elsevier