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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
##### Citations:
Zbl 1231.11038; Zbl 1362.11045
OEIS
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##### References:
 [1] Chowla, S.; Hartung, P., An “exact” formula for the m-th Bernoulli number, Acta arith., 22, 113-115, (1972) · Zbl 0244.10008 [2] Gallot, Y.; Moree, P.; Zudilin, W., The Erdős-Moser equation $$1^k + 2^k + \cdots +(m - 1)^k = m^k$$ revisited using continued fractions, Math. comp., 80, 1221-1237, (2011) · Zbl 1231.11038 [3] Graham, R.L.; Knuth, D.E.; Patashnik, O., Concrete mathematics, (1994), Addison-Wesley Reading, MA, USA · Zbl 0836.00001 [4] Ireland, K.; Rosen, M., A classical introduction to modern number theory, Grad. texts in math., vol. 84, (1990), Springer-Verlag · Zbl 0712.11001 [5] Kellner, B.C., On irregular prime power divisors of the Bernoulli numbers, Math. comp., 76, 405-441, (2007) · Zbl 1183.11012 [6] P. Moree, H.J.J. te Riele, J. Urbanowicz, Divisibility properties of integers x and k satisfying 1k+2k+⋯+(x−1)k=xk, CWI Reports and Notes, Numerical Mathematics, 1992. [7] Moser, L., On the Diophantine equation $$1^n + 2^n + 3^n + \cdots +(m - 1)^n = m^n$$, Scripta math., 19, 84-88, (1953) · Zbl 0050.26604 [8] Sloane, N.J.A., Online encyclopedia of integer sequences (OEIS), electronically published at: · Zbl 1274.11001
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