Divisibility properties of integers 𝑥,𝑘 satisfying 1^{𝑘}+⋯+(𝑥-1)^{𝑘}=𝑥^{𝑘}

Moree, P.; te Riele, H. J. J.; Urbanowicz, J.

doi:10.1090/S0025-5718-1994-1257577-1

Divisibility properties of integers $x,\ k$ satisfying $1^ k+\cdots +(x-1)^ k=x^ k$
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by P. Moree, H. J. J. te Riele and J. Urbanowicz PDF

Math. Comp. 63 (1994), 799-815 Request permission

Abstract:

Based on congruences $\bmod \;p$ and on properties of Bernoulli polynomials and Bernoulli numbers, several conditions are derived for x, $x,k \geq 2$ to satisfy the Diophantine equation ${1^k} + {2^k} + \cdots + {(x - 1)^k} = {x^k}$. It is proved that ${\text {ord}_2}(x - 3) = {\text {ord}_2}k + 3$ and that x cannot be divisible by any regular prime. Furthermore, by using the results of experiments with the above conditions on an SGI workstation it is proved that x cannot be divisible by any irregular prime $< 10000$ and that k is divisible by the least common multiple of all the integers $\leq 200$.

References

On a conjecture of Erdös concerning sums of powers of integers

J. P. Buhler, R. E. Crandall, and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 (1992), no. 200, 717–722. MR 1134717, DOI 10.1090/S0025-5718-1992-1134717-4
H. Davenport and D. J. Lewis, Homogeneous additive equations, Proc. Roy. Soc. London Ser. A 274 (1963), 443–460. MR 153655, DOI 10.1098/rspa.1963.0143
Hubert Delange, Sur les zéros réels des polynômes de Bernoulli, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 2, 267–309 (French, with English summary). MR 1137289
Richard K. Guy, Unsolved problems in number theory, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1981. MR 656313
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
Wells Johnson, On the vanishing of the Iwasawa invariant $\mu _{p}$ for $p<8000$, Math. Comp. 27 (1973), 387–396. MR 384748, DOI 10.1090/S0025-5718-1973-0384748-5
Wells Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653–657. MR 347727, DOI 10.1090/S0025-5718-1974-0347727-0

A note on the equation

On a conjecture of Erdös

Divisibility properties of integers x and k satisfying

Leo Moser, On the diophantine equation $1^n+2^n+3^n+\cdots +(m-1)^n=m^n.$, Scripta Math. 19 (1953), 84–88. MR 54627
Paulo Ribenboim, 13 lectures on Fermat’s last theorem, Springer-Verlag, New York-Heidelberg, 1979. MR 551363
Harold N. Shapiro, Introduction to the theory of numbers, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 693458
Jerzy Urbanowicz, Remarks on the equation $1^k+2^k+\cdots +(x-1)^k=x^k$, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 3, 343–348. MR 964840
H. S. Vandiver, On Bernoulli’s numbers and Fermat’s last theorem, Duke Math. J. 3 (1937), no. 4, 569–584. MR 1546011, DOI 10.1215/S0012-7094-37-00345-4
Samuel S. Wagstaff Jr., The irregular primes to $125000$, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465, DOI 10.1090/S0025-5718-1978-0491465-4
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Guo Fu Zhou and Chi Ding Kang, On the Diophantine equation $\sum ^{m}_{k=1}k^{n}=(m+1)^{n}$, J. Math. Res. Exposition 3 (1983), no. 4, 47–48. MR 744117