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On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-\(p\) digits. (English) Zbl 07354559
Summary: We give a new characterization of the set \(\mathcal{C}\) of Carmichael numbers in the context of \(p\)-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-\(p\)-digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a \(p\)-adically defined subset \(\mathcal{S}\) of the squarefree integers that contains \(\mathcal{C}\). This leads to the definition of a new subset \(\mathcal{C}^{\prime}\) of \(\mathcal{C}\), called the “primary Carmichael numbers”. Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set \(\mathcal{S}\) is covered by modular subsets \(\mathcal{S}_d\) \((d \ge 1)\) that are related to the Knödel numbers, where \(\mathcal{C}=\mathcal{S}_1\) is a special case.
11A51 Factorization; primality
11B68 Bernoulli and Euler numbers and polynomials
11A63 Radix representation; digital problems
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[1] W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math.139(1994), 703-722. · Zbl 0816.11005
[2] A. H. Beiler,Recreations in the Theory of Numbers, Dover, New York, 1966. · Zbl 0154.04001
[3] R. D. Carmichael, Note on a new number theory function,Bull. Amer. Math. Soc.16(1910), 232-238. · JFM 41.0226.04
[4] R. D. Carmichael, On composite numbersPwhich satisfy the Fermat congruenceaP−1≡ 1 (modP),Amer. Math. Monthly19(1912), 22-27.
[5] T. Clausen, Lehrsatz aus einer Abhandlung ¨uber die Bernoullischen Zahlen,Astr. Nachr.17 (1840), 351-352.
[6] K. Conrad, Carmichael numbers and Korselt’s criterion, expository paper (2016), 1-3. Availableathttp://www.math.uconn.edu/ kconrad/blurbs/ugradnumthy/carmichaelkorselt. pdf.
[7] J. H. Conway and R. K. Guy,The Book of Numbers, Springer-Verlag, New York, 1996. · Zbl 0866.00001
[8] R. Crandall and C. B. Pomerance,Prime Numbers: A Computational Perspective, 2nd ed., Springer, New York, 2005. · Zbl 1088.11001
[9] P. Erd˝os, On pseudoprimes and Carmichael numbers,Publ. Math. Debrecen4(1956), 201-206.
[10] A. Granville and C. Pomerance, Two contradictory conjectures concerning Carmichael numbers,Math. Comp.71(2002), 883-908. · Zbl 0991.11067
[11] R. K. Guy,Unsolved Problems in Number Theory, 3rd ed., Springer, New York, 2004. · Zbl 1058.11001
[12] G. H. Hardy,Ramanujan, Cambridge Univ. Press, New York, 1940.
[13] G. Harman, Watt’s mean value theorem and Carmichael numbers,Int. J. Number Theory4 (2008), 241-248. · Zbl 1221.11194
[14] D. R. Heath-Brown, Carmichael numbers with three prime factors,Hardy-Ramanujan J.30 (2007), 6-12. · Zbl 1151.11047
[15] B. C. Kellner, On a product of certain primes,J. Number Theory179(2017), 126-141. · Zbl 1418.11045
[16] B. C. Kellner and J. Sondow, Power-sum denominators,Amer. Math. Monthly124(2017), 695-709. · Zbl 1391.11052
[17] B. C. Kellner and J. Sondow, The denominators of power sums of arithmetic progressions, Integers18(2018), #A95, 1-17. · Zbl 1423.11029
[18] W. Kn¨odel, Carmichaelsche Zahlen,Math. Nachr.9(1953), 343-350.
[19] W. Kn¨odel, Eine obere Schranke f¨ur die Anzahl der Carmichaelschen Zahlen kleiner alsx, Arch. Math.4(1953), 282-284.
[20] A. Korselt, Probl‘eme chinois,L’Interm´ediaire Math.6(1899), 142-143.
[21] A. Makowski, Generalization of Morrow’sDnumbers,Simon Stevin36(1962), 71.
[22] R. G. E. Pinch, The Carmichael numbers up to 1021, Proceedings of Conference on Algorithmic Number Theory 2007, A. Ernvall-Hyt¨onen et al., eds.,TUCS General Publication46, Turku Centre for Computer Science, 2007, 129-131.
[23] C. Pomerance, J. L. Selfridge, and S. Wagstaff, The pseudoprimes to 25·109,Math. Comp. 35(1980), 1003-1026. · Zbl 0444.10007
[24] P. Ribenboim,The New Book of Prime Number Records, Springer, New York, 2012. · Zbl 0856.11001
[25] A. M. Robert,A Course inp-adic Analysis, GTM198, Springer-Verlag, New York, 2000.
[26] K. G. C. von Staudt, Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend,J. Reine Angew. Math.21(1840), 372-374
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