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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.
##### MSC:
 11A51 Factorization; primality 11B68 Bernoulli and Euler numbers and polynomials 11A63 Radix representation; digital problems
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##### References:
 [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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