Andrejić, Vladica; Bostan, Alin; Tatarevic, Milos Improved algorithms for left factorial residues. (English) Zbl 1515.11123 Inf. Process. Lett. 167, Article ID 106078, 4 p. (2021). Summary: We present improved algorithms for computing the left factorial residues \(!p=0!+1!+\dots+(p-1)!\bmod p\). We use these algorithms for the calculation of the residues \(!p\bmod p\), for all primes \(p\) up to \(2^{40}\). Our results confirm that Kurepa’s left factorial conjecture is still an open problem, as they show that there are no odd primes \(p<2^{40}\) such that \(p\) divides !\(p\). Additionally, we confirm that there are no socialist primes p with \(5<p<2^{40}\). Cited in 3 Documents MSC: 11Y55 Calculation of integer sequences 11B65 Binomial coefficients; factorials; \(q\)-identities 68W30 Symbolic computation and algebraic computation Keywords:Kurepa’s conjecture; left factorial; socialist primes; fast algorithms; software design and implementation Software:NTL; FLINT; gmp PDFBibTeX XMLCite \textit{V. Andrejić} et al., Inf. Process. Lett. 167, Article ID 106078, 4 p. (2021; Zbl 1515.11123) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: a(n) = (0! + 1! + ... + (p-1)!) mod p, where p = prime(n). Calculate the smallest positive residues of 2!,3!,...,(prime(n)-1)! modulo prime(n) and stop the calculation at the moment m when for the first time there appear two equal numbers. If m!==k!, then a(n)=k; but a(n)=0 if no such m exists. Prime numbers p such that K(p) = 0! + 1! + ... + (p-1)! == -2 (mod p). References: [1] V. Andrejić, On Kurepa’s left factorial conjecture, in: XIV Serbian Mathematical Congress, May 16-19, 2018, Kragujevac, Serbia, Book of abstracts, p. 96. [2] Andrejić, V.; Tatarevic, M., Searching for a counterexample to Kurepa’s conjecture, Math. Comput., 85, 3061-3068 (2016) · Zbl 1360.11002 [3] Andrejić, V.; Tatarevic, M., On distinct residues of factorials, Publ. Inst. Math. (N.S.), 100, 101-106 (2016) · Zbl 1432.11016 [4] Bostan, A.; Gaudry, P.; Schost, E., Linear recurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator, SIAM J. Comput., 36, 1777-1806 (2007) · Zbl 1210.11126 [5] Costa, E.; Gerbicz, R.; Harvey, D., A search for Wilson primes, Math. Comput., 83, 3071-3091 (2014) · Zbl 1370.11003 [6] Granlund, T.; the GMP development team, GNU MP: the GNU multiple precision arithmetic library (2016), Version 6.1.2 [7] Guy, R., Unsolved Problems in Number Theory (2004), Springer-Verlag · Zbl 1058.11001 [8] Hart, W.; Johansson, F.; Pancratz, S., FLINT: fast library for number theory (2015), Version 2.5.2 [9] Harvey, D., NTT: a library for large integer arithmetic (2012), Version 0.1.2 [10] Kurepa, Đ., On the left factorial function !n, Math. Balk., 1, 147-153 (1971) · Zbl 0224.10009 [11] Rajkumar, R., Searching for a counterexample to Kurepa’s conjecture in average polynomial time (2019), School of Mathematics and Statistics, UNSW Sydney, Master’s thesis [12] Shoup, V., NTL: a library for doing number theory (2016), Version 10.3.0 [13] Trudgian, T., There are no socialist primes less than 10^9, Integers, 14 (2014), #A63 · Zbl 1336.11009 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.