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Divisibility of power sums and the generalized Erdős-Moser equation. (English) Zbl 1264.11025
For $k$ an integer, let $\nu_2(k)$ denote the highest exponent $\nu$ such that $2^{\nu}$ divides $k$. Given positive integers $m$ and $n$ the authors relate, using induction, $\nu_2(1^n+2^n+\cdots+m^n)$ to $\nu_2(m(m+1)/2)$. They apply this result to give an easy reproof of the result of the reviewer [Bull. Aust. Math. Soc. 53, No. 2, 281--292 (1996; Zbl 0851.11020)] that if $1^n+2^n+\cdots+(m-1)^n=am^n$, then $m$ must be odd.

MSC:
11D41Higher degree diophantine equations
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References:
[1] Dickson, L.E.: History of the Theory of Numbers. Volume II: Diophantine Analysis . Carnegie Institute, Washington, D.C., 1919; reprinted by Dover, New York 2005.
[2] Edwards, A.W.F.: A quick route to sums of powers. Amer. Math. Monthly 93 (1986), 451-455. · Zbl 0605.40004 · doi:10.2307/2323466
[3] Guy, R.K.: Unsolved Problems in Number Theory . 3rd ed. Springer, New York 2004. · Zbl 1058.11001
[4] Hayes, B.: Gauss’s day of reckoning. Amer. Scientist 94 (2006), 200-205; see http://www.sigmaxi.org/amscionline/gauss-snippets.html.
[5] Lengyel, T.: On divisibility of some power sums. Integers 7 (2007) A41, 1-6. · Zbl 1132.11302 · eudml:128503
[6] Lengyel, T.: Personal communication. 4 November 2010.
[7] MacMillan, K.; Sondow, J.: Reducing the Erd\Acute\Acute os-Moser equation 1n + 2n + \cdot \cdot \cdot + kn = (k + 1)n modulo k and k2. Integers 11 (2011) A34, 1-8.
[8] Moree, P.: Diophantine equations of Erd\Acute\Acute os-Moser type. Bull. Austral. Math. Soc. 53 (1996), 281-292. · Zbl 0851.11020 · doi:10.1017/S0004972700017007
[9] Moree, P.: Moser’s mathemagical work on the equation 1k + 2k + \cdot \cdot \cdot + (m - 1)k = mk. Rocky Mountain J. Math. (to appear); available at http://arxiv.org/abs/1011.2940
[10] Moree, P.; Te Riele, H.; Urbanowicz, J.: Divisibility properties of integers x, k satisfying 1k + 2k + \cdot \cdot \cdot + (x - 1)k = xk. Math. Comp. 63 (1994), 799-815. · Zbl 0816.11024 · doi:10.2307/2153300
[11] Moser, L.: On the Diophantine equation 1n + 2n + \cdot \cdot \cdot + (m - 1)n = mn. Scripta Math. 19 (1953), 84-88.