On the number of solutions of the equation \(1^k+2^k+\dots+(x-1)^k=y^z\) (Q2707218)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the number of solutions of the equation \(1^k+2^k+\dots+(x-1)^k=y^z\) |
scientific article |
Statements
1 April 2001
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Bernoulli polynomials
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counting solutions of diophantine equations
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sum of powers
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On the number of solutions of the equation \(1^k+2^k+\dots+(x-1)^k=y^z\) (English)
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For a positive integer \(k\) let \(S_k(x)=1^k+2^k+\ldots +(x-1)^k\). In the paper the following two theorems are proved. Theorem 1. The equation \(S_k(x)=y^z\) has at most \(\max(c_1, e^{3k})\) solutions in positive integers \(x, y>1, \;z>2\) and \((k, z)\neq (3, 4) \), where \(c_1\) is an effectively computable absolute constant. Theorem 2. If \(k\) is even then the equation \(S_k(x)=y^2\) possesses at most \(\max(c_2, 9^k)\) solutions in positive integers \(x\) and \(y\), where \(c_2\) is an effectively computable absolute constant.
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