×

zbMATH — the first resource for mathematics

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

MSC:
11D61 Exponential Diophantine equations
11Y65 Continued fraction calculations (number-theoretic aspects)
11A55 Continued fractions
PDF BibTeX XML Cite
Full Text: DOI arXiv
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/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.