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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.

11D61 Exponential Diophantine equations
11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials
Full Text: DOI arXiv
[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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