Bell numbers and Kurepa’s conjecture

Luis Gallardo

Abstract


We prove under a mild condition that Kurepa's conjecture holds for the set of prime numbers \(p\) such that \((\frac{p-1}{2})! = {2 \overwithdelims () p\;}\) in \(\mathbb{F}_p\).

Keywords


Artin–Schreier extension; Bell numbers; Kurepa conjecture

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References


Aigner, M., A characterization of the Bell numbers, Discrete Math. 205(1–3) (1999), 207–210.

Andrejic, V., Bostan, A., Tatarevic, M., On distinct residues of factorials, Publ. Inst. Math. Nouv. Ser. 100(114) (2016), 101–106.

Andrejic, V., Tatarevic, M., Searching for a counterexample to Kurepa’s conjecture, Math. Comput. 85(302) (2016), 3061–3068.

Andrejic, V., Bostan, A., Tatarevic, M., Improved algorithms for left factorial residues, Inf. Process. Lett. 167 (2021), Article ID 106078, 4 pp.

Barsky, D., Benzaghou, B., Nombres de Bell et somme de factorielles, J. Theor. Nombres Bordeaux 16(1) (2004), 1–17.

Barsky, D., Benzaghou, B., Erratum a l’article Nombres de Bell et somme de factorielles, J. Theor. Nombres Bordeaux 23(2) (2011), 527.

Becker, H. W., Riordan, J., The arithmetic of Bell and Stirling numbers, Amer. J. Math. 70 (1948), 385–394.

Carlitz, L., A note on the left factorial function, Math. Balkanica 5 (1975), 37–42.

Dalton, R. E., Levine, J., Minimum periods, modulo p, of first order Bell exponential integers, Math. Comp. 16 (1962), 416–423.

d’Ocagne, M., Sur une classe de nombres remarquables, Amer. J. Math. 9 (1887), 353–380.

Dragovic, B., On some finite sums with factorials, Facta Univ., Ser. Math. Inf. 14 (1999), 1–10.

Gallardo, L. H., Rahavandrainy, O., Bell numbers modulo a prime number, traces and trinomials, Electron. J. Comb. 21(4) (2014), Research Paper P4.49, 30 pp.

Graham, R. L., Knuth, D. E., Patashnik, O., Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989.

Ivic, A., Mijajlovic, ˘Z, On Kurepa’s problems in number theory, Publ. Inst. Math. (Beograd) (N.S.), Duro Kurepa memorial volume, 57(71) (1995), 19–28.

Kahale, N., New modular properties of Bell numbers, J. Combin. Theory Ser. A 58(1) (1991), 147–152.

Kohnen, W., A remark on the left-factorial hypothesis, Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. 9 (1998), 51–53.

Kurepa, D., On the left factorial function !n, Math. Balkanica 1(1) (1971), 147–153.

Kurepa, D., Right and left factorials, in: Conferenze tenuti in occasione del cinquantenario dell’ Unione Matematica Italiana (1972), Boll. Un. Mat. Ital 4(9) (1971), 171–189.

Kurepa, D., On some new left factorial propositions, Math. Balkanica 4 (1974), 383–386.

Lidl, R., Niederreiter, H., Finite Fields, Encyclopedia of Mathematics and its applications, Cambridge University Press (1983), Reprinted, 1987.

Mijajlovic, Z., On some formulas involving !n and the verification of the !n-hypothesis by use of computers, Publ. Inst. Math. (Beograd) (N.S.) 47(61) (1990), 24–32.

Montgomery, P., Nahm, S.,Wagstaff, Jr., S. S., The period of the Bell numbers modulo a prime, Math. Comp. 79(271) (2010), 1793–1800.

Petojevic, A., On Kurepa’s hypothesis for the left factorial, Filomat 12(1) (1998), 29–37.

Petojevic, A., Zizovic, M., Trees and the Kurepa hypothesis for left factorial, Filomat 13 (1999), 31–40.

Petojevic, A., Zizovic, M., Cvejic, S. D., Difference equations and new equivalents of the Kurepa hypothesis, Math. Morav. 3 (1999), 39–42.

Radoux, Chr., Determinants de Hankel et theoreme de Sylvester, in: Seminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), 115–122, Publ. Inst. Rech. Math. Av., 498, Univ. Louis Pasteur, Strasbourg, 1992.

Sami, Z., On generalization of functions n! and !n, Publ. Inst. Math., Nouv. Ser. 60(74) (1996), 5–14.

Sami, Z., A sequence un,m and Kurepa’s hypothesis on left factorial, in: Symposium Dedicated to the Memory of Duro Kurepa (Belgrade, 1996), Sci. Rev. Ser. Sci. Eng. 19–20 (1996), 125–113.

Sloane, N. J. A., et al., The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2019.

Stankovic, J., Uber einige Relationen zwischen Fakultaten und den linken Fakultaten, Math. Balkanica 3 (1973), 488–495.

Stankovic, J., Zizovic, M., Noch einige Relationen zwischen den Fakultaten und den linken Fakultaten, Math. Balkanica 4 (1974), 555–559.

Trudgian, T., There are no socialist primes less than 109, Integers 14 (2014), Paper A63, 4 pp.

Vladimirov, V. S., Left factorials, Bernoulli numbers, and the Kurepa conjecture, Publ. Inst. Math. (Beograd) (N.S.) 72(86) (2002), 11–22.

Williams, G. T., Numbers generated by the function eex-1, Amer. Math. Monthly 52 (1945), 323–327.




DOI: http://dx.doi.org/10.17951/a.2022.76.1.17-23
Date of publication: 2022-10-05 20:39:31
Date of submission: 2022-10-04 18:57:13


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