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## Section: New Results

### Improved algorithms for left factorial residues

In [11], Alin Bostan together with Vladica Andrejić (University of Belgrade, Serbia) and Milos Tatarevic (CoinList, Alameda, CA) presented improved algorithms for computing the left factorial residues $!p=0!+1!+\cdots +\left(p-1\right)!\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}p$. They used these algorithms for the calculation of the residues $!p\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}p$, for all primes $p$ up to ${2}^{40}$. Their 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, they confirmed that there are no socialist primes $p$ with $5.