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On stronger conjectures that imply the Erdős-Moser conjecture. (English) Zbl 1267.11031
The Erdős-Moser conjecture states that the Diophantine equation \(S_k(m)=m^k\), where \(S_k(m)=1^k+2^k+\cdots+(m-1)^k\) has no solution for \(m>3\). The best result to date is that it has no solution with \(m>10^{10^9}\), due to Y. Gallot, the reviewer and W. Zudilin [Math. Comput. 80, No. 274, 1221–1237 (2011; Zbl 1231.11038)]. Here the author states the conjecture that \(S_k(m+1)/S_k(m)\) is never an integer for \(m>3\). This is equivalent with the statement that the Diophantine equation \(aS_k(m)=m^k\) has no solution with \(m>3\) and \(a\geq 1\) an integer. Meanwhile the reviewer [P. Moree, “Moser’s mathemagical work on the equation \(1^k+2^k+\cdots+(m-1)^k=m^k\)”, arXiv:1011.2940, Rocky Mt. J. Math. 43, No. 5, 1707–1737 (2013; Zbl 1362.11045)] has shown that there are infinitely many \(a\) for which this equation has no solution.
Let \(g_k(m)\) be the greatest common divisor of \(S_k(m)\) and \(S_k(m+1)\) divided by \(m\). Let \(N_k\) denote the numerator of the \(k\)th Bernoulli number. The author proves that if \[ \max_{m\geq 1}g_k(m)< |N_k|(\log |N_k|)^6 \] for \(k\geq 10\), then the Erdős-Moser conjecture holds true and that more generally the exponent 6 can be replaced by an arbitrary exponent \(e\), at cost of also requiring \(k\geq C_e\), where \(C_e\) is effectively computable. Furthermore, he conjectures that the latter upper bound for \(\max_{m\geq 1}g_k(m)\) holds true for some exponent \(e\) and established this in case \(N_k\) is square-free by showing that \(\max_{m\geq 1}g_k(m)=|N_k|\) then.

MSC:
11D61 Exponential Diophantine equations
11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials
Software:
OEIS
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