## 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

### Citations:

Zbl 1231.11038; Zbl 1362.11045

OEIS
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### References:

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