The Erdős–Moser equation \(1^k+2^k+\cdots +(m-1)^k=m^k\) revisited using continued fractions. (English) Zbl 1231.11038

Gallot, Yves; Moree, Pieter; Zudilin, Wadim

Math. Comput. 80, No. 274, 1221-1237 (2011).

In the paper the authors investigate the non-trivial solutions of the equation \[ 1^k+2^k+\cdots +(m-2)^k+(m-1)^k=m^k \] with \(k\geq 2\). Conjecturally such solutions do not exist. L. Moser [Scripta Math. 19, 84–88 (1953; Zbl 0050.26604)] showed that if \((m,k)\) is a solution of this equation then \(m>10^{10^6}\) and \(k\) is even. Using the method of Moser this result cannot be improved on substantially. Now the authors prove that \(m>10^{10^9}\) using an extensive continued fraction digits calculation of \(\frac{\log 2}{N}\) where \(N\) is an appropriate integer. The new method uses a large scale computation of a numerical constant.

Reviewer: Kálmán Liptai (Eger)

MSC:

11D61	Exponential Diophantine equations
11Y65	Continued fraction calculations (number-theoretic aspects)
11A55	Continued fractions

Keywords:

continued fraction; exponential equation; rational approximation

Citations:

Zbl 0050.26604

References:

[1]	Horst Alzer, Über eine Verallgemeinerung der Bernoullischen Ungleichung, Elem. Math. 45 (1990), no. 2, 53 – 54 (German). · Zbl 0723.26008
[2]	Jonathan Borwein and David Bailey, Mathematics by experiment, 2nd ed., A K Peters, Ltd., Wellesley, MA, 2008. Plausible reasoning in the 21st Century. · Zbl 1163.00002
[3]	M. R. BEST and H. J. J. TE RIELE, On a conjecture of Erdős concerning sums of powers of integers, Report NW 23/76 (Mathematisch Centrum Amsterdam, 1976). · Zbl 0326.10018
[4]	R. P. BRENT, A. J. VAN DER POORTEN, and H. TE RIELE, A comparative study of algorithms for computing continued fractions of algebraic numbers, Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci. 1122 (Springer, Berlin, 1996), 35-47. · Zbl 0899.11065
[5]	Lawrence Brenton and Ana Vasiliu, Znam’s problem, Math. Mag. 75 (2002), no. 1, 3 – 11.
[6]	William Butske, Lynda M. Jaje, and Daniel R. Mayernik, On the equation \sum _{\?|\?}(1/\?)+(1/\?)=1, pseudoperfect numbers, and perfectly weighted graphs, Math. Comp. 69 (2000), no. 229, 407 – 420. · Zbl 0934.11015
[7]	L. Carlitz, The Staudt-Clausen theorem, Math. Mag. 34 (1960/1961), 131 – 146. · Zbl 0122.04702
[8]	Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071
[9]	D. R. Curtiss, On Kellogg’s Diophantine Problem, Amer. Math. Monthly 29 (1922), no. 10, 380 – 387. · JFM 48.0157.02
[10]	Hubert Delange, Sur les zéros réels des polynômes de Bernoulli, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 2, 267 – 309 (French, with English summary). · Zbl 0725.11011
[11]	P. ERDŐS, Advanced Problem 4347, Amer. Math. Monthly 56 (1949), 343.
[12]	Leon Gerber, An extension of Bernoulli’s inequality, Amer. Math. Monthly 75 (1968), 875 – 876. · Zbl 0165.37101
[13]	Daniel B. Grünberg and Pieter Moree, Sequences of enumerative geometry: congruences and asymptotics, Experiment. Math. 17 (2008), no. 4, 409 – 426. With an appendix by Don Zagier. · Zbl 1182.11047
[14]	Richard K. Guy, Unsolved problems in number theory, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. · Zbl 1058.11001
[15]	Glyn Harman and Kam C. Wong, A note on the metrical theory of continued fractions, Amer. Math. Monthly 107 (2000), no. 9, 834 – 837. · Zbl 0982.11043
[16]	Christopher Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209 – 220. · Zbl 0221.10048
[17]	Hendrik Jager and Pierre Liardet, Distributions arithmétiques des dénominateurs de convergents de fractions continues, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 2, 181 – 197 (French). · Zbl 0655.10045
[18]	Y. KANADA, Kanada \( \pi\)-Laboratory, available at http://www.super-computing.org/.
[19]	B. C. KELLNER, Über irreguläre Paare höhere Ordnungen, Diplomarbeit (Mathematisches Institut der Georg-August-Universität zu Göttingen, Germany, 2002); available at http://www.bernoulli.org/\~bk/irrpairord.pdf.
[20]	Bronisław Krzysztofek, The equation 1\(^{n}\)+2\(^{n}\)+\cdots+\?\(^{n}\)=(\?+1)\(^{n}\)\cdot \?, Wyż. Szkoł. Ped. w Katowicach — Zeszyty Nauk. Sekc. Mat. 5 (1966), 47 – 54 (1967) (Polish, with English summary).
[21]	Paul Lévy, Sur le développement en fraction continue d’un nombre choisi au hasard, Compositio Math. 3 (1936), 286 – 303 (French). · Zbl 0014.26803
[22]	P. LIARDET, Propriétés arithmétiques presque sûres des convergents, Séminaire de théorie des Nombres de Bordeaux (1986-87), exp. no. 36, 20 pp.
[23]	R. Moeckel, Geodesics on modular surfaces and continued fractions, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 69 – 83. · Zbl 0497.10007
[24]	Niels Möller, On Schönhage’s algorithm and subquadratic integer GCD computation, Math. Comp. 77 (2008), no. 261, 589 – 607. · Zbl 1165.11003
[25]	Pieter Moree, On a theorem of Carlitz-von Staudt, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), no. 4, 166 – 170. · Zbl 0820.11002
[26]	Pieter Moree, Diophantine equations of Erdős-Moser type, Bull. Austral. Math. Soc. 53 (1996), no. 2, 281 – 292. · Zbl 0851.11020
[27]	P. Moree, H. J. J. te Riele, and J. Urbanowicz, Divisibility properties of integers \?,\? satisfying 1^{\?}+\cdots+(\?-1)^{\?}=\?^{\?}, Math. Comp. 63 (1994), no. 208, 799 – 815. · Zbl 0816.11024
[28]	P. MOREE, A top hat for Moser’s four mathemagical rabbits, Amer. Math. Monthly (to appear). · Zbl 1228.11045
[29]	Leo Moser, On the diophantine equation 1\(^{n}\)+2\(^{n}\)+3\(^{n}\)+\cdots+(\?-1)\(^{n}\)=\?\(^{n}\)., Scripta Math. 19 (1953), 84 – 88. · Zbl 0050.26604
[30]	R. W. K. Odoni, On the prime divisors of the sequence \?_{\?+1}=1+\?\(_{1}\)\cdots\?_{\?}, J. London Math. Soc. (2) 32 (1985), no. 1, 1 – 11. · Zbl 0574.10020
[31]	Jerzy Urbanowicz, Remarks on the equation 1^{\?}+2^{\?}+\cdots+(\?-1)^{\?}=\?^{\?}, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 3, 343 – 348. · Zbl 0661.10025
[32]	A. J. YEE, \( \gamma\)-cruncher–A Multi-Threaded Pi-Program, see http://www.numberworld.org/.