zbMATH — the first resource for mathematics

Power-sum denominators. (English) Zbl 1391.11052
Summary: The power sum \(1^n+2^n+\dots+x^n\) has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in \(x\) of degree \(n+1\) with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals \(n+1\) times the squarefree product of certain primes \(p\) obeying the condition that the sum of the base-\(p\) digits of \(n+1\) is at least \(p\). As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

11B83 Special sequences and polynomials
11B68 Bernoulli and Euler numbers and polynomials
Full Text: DOI
[1] G. Almkvist, A. Meurman, Values of Bernoulli polynomials and Hurwitz’s zeta function at rational points, C. R. Math. Acad. Sci. Soc. R. Can. 13 no. 2-3 (1991) 104-108. · Zbl 0731.11014
[2] P. Bachmann, Niedere Zahlentheorie. Part 2, Teubner, Leipzig, 1910; Parts 1 and 2 reprinted in one volume, Chelsea, New York, 1968, http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN379887479.
[3] J. Beery, Sums of Powers of Positive Integers — Johann Faulhaber (1580-1635), Germany, Convergence (July 2010), http://www.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-johann-faulhaber-1580-1635-germany.
[4] J. Bernoulli, Ars Conjectandi.Basel, 1713, http://dx.doi.org/10.3931/e-rara-9001.
[5] T. Clausen, Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840) 351-352.
[6] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics. Vol. 240, Springer-Verlag, New York, 2007. · Zbl 1119.11002
[7] J. H. Conway, R. K. Guy, The Book of Numbers.Springer-Verlag, New York, 1996. · Zbl 0866.00001
[8] A. W. F. Edwards, A quick route to sums of powers, Amer. Math. Monthly 9 (1986) 451-455. · Zbl 0605.40004
[9] L. Euler, Inventio summae cuiusque seriei ex dato termino generali, E47, Comment. Acad. Sc. Petrop. 8 (1741) 9-22, http://eulerarchive.maa.org/pages/E047.html.
[10] J. Faulhaber, Newer Arithmetischer Wegweyser.Johann Meder, Ulm, 1614.
[11] J. Faulhaber, Academia Algebrae.Johann Remmelin, Augsburg, 1631, http://dx.doi.org/10.3931/e-rara-16627.
[12] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947) 589-592. · Zbl 0030.11102
[13] S. Furck, Portrait of Johann Faulhaber.Copperplate engraving, about 1630, Stadtarchiv Ulm, F 4 Bildnis 100, http://www.stadtarchiv-ulm.findbuch.net/php/main.php?ar_id=3766#4620342042696c646e69737365x252.
[14] J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quart. J. PureAppl. Math. 30 (1899) 150-156. · JFM 29.0152.03
[15] J. V. Grabiner, Was Newton’s calculus a dead end? The continental influence of Maclaurin’s Treatise of Fluxions, Amer. Math. Monthly 104 (1997) 393-410. · Zbl 0885.01010
[16] A. Granville, ZaphodBeeblebrox’s brain and the fifty-ninth row of Pascal’s triangle, Amer. Math. Monthly 99 (1992) 318-331. Correction, Amer. Math. Monthly 104 (1997) 848-851. · Zbl 0757.05003
[17] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers.Fifth ed.Oxford Univ. Press, Oxford, 1989. · Zbl 0020.29201
[18] C. Hermite, Extrait d’une lettre a M. Borchardt, J. Reine Angew. Math. 81 (1876) 93-95, http://www.digizeitschriften.de/dms/img/?PID=PPN243919689_0081/log8.
[19] D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993) 277-294, http://dx.doi.org/10.1090/S0025-5718-1993-1197512-7. · Zbl 0797.11026
[20] A.-M. Legendre, Essai sur la Théorie des Nombres.Second ed.Courcier, Paris, 1808, http://gallica.bnf.fr/ark:/12148/bpt6k62826k/f37.
[21] É. Lucas, Théorie des Nombres.Gauthier-Villars, Paris, 1891.
[22] C. Maclaurin, A Treatise of Fluxions in Two Books.Ruddimans, Edinburgh, 1742.
[23] K. MacMillan, J. Sondow, Proofs of power sum and binomial coefficient congruences via Pascal’s identity, Amer. Math. Monthly 118 (2011) 549-551. · Zbl 1230.05014
[24] V. H. Moll, Numbers and Functions: From a Classical-Experimental Mathematician’s Point of View. Student Mathematical Library. Vol. 65, American Mathematical Society, Providence, RI, 2012. · Zbl 1268.00011
[25] J. J. O’Connor, E. F. Robertson, Biography of Johann Faulhaber in MacTutor History of Mathematics, http://www-history.mcs.st-and.ac.uk/Biographies/Faulhaber.html.
[26] I. Schneider, Potenzsummenformeln im 17. Jahrhundert, Hist. Math. 10 (1983) 286-296, http://dx.doi.org/10.1016/0315-0860(83)90079-4. · Zbl 0522.01004
[27] I. Schneider, Johannes Faulhaber 1580-1635, Rechenmeister in einer Welt des Umbruchs. Vita Mathematica 7, Birkhäuser-Verlag, Basel, 1993. · Zbl 0786.01020
[28] W. Sierpiński, Sur une courbe dont tout point est un point de ramification, C. R. Acad. Sci. Paris 160 (1915) 302-305. · JFM 45.0628.02
[29] N. J. A. Sloane, ed., The On-Line Encyclopedia of Integer Sequences, http://oeis.org. · Zbl 1274.11001
[30] K. G. C. von Staudt, Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend, J. Reine Angew. Math. 21 (1840) 372-374, http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002142562.
[31] J. Stopple, A Primer of Analytic Number Theory: From Pythagoras to Riemann.Cambridge Univ. Press, Cambridge, 2003. · Zbl 1029.11001
[32] F. J. Swetz, V. J. Katz, Mathematical Treasures — Johann Faulhaber’s Academia Algebrae, Convergence (January 2011), http://www.maa.org/press/periodicals/convergence/mathematical-trea-sures-johann-faulhabers-academia-algebrae.
[33] S. Wolfram, Geometry of binomial coefficients, Amer. Math. Monthly 91 (1984) 566-571. · Zbl 0553.05004
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.