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Abstract We report that there are 49679870 Carmichael numbers less than$$10^{22}$$ ${10}^{22}$which is an order of magnitude improvement on Richard Pinch’s prior work. We find Carmichael numbers of the form$$n = Pqr$$ $n=Pqr$using an algorithm bifurcated by the size ofPwith respect to the tabulation boundB. For$$P < 7 \times 10^7$$ $P<7\times {10}^7$, we found 35985331 Carmichael numbers and 1202914 of them were less than$$10^{22}$$ ${10}^{22}$. When$$P > 7 \times 10^7$$ $P>7\times {10}^7$, we found 48476956 Carmichael numbers less than$$10^{22}$$ ${10}^{22}$. We provide a comprehensive overview of both cases of the algorithm. For the large case, we show and implement asymptotically faster ways to tabulate compared to the prior tabulation. We also provide an asymptotic estimate of the cost of this algorithm. It is interesting that Carmichael numbers are worst case inputs to this algorithm. So, providing a more robust asymptotic analysis of the cost of the algorithm would likely require resolution of long-standing open questions regarding the asymptotic density of Carmichael numbers. 

Abstract We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integern, there exists a prime between$$n^2$$ $n^2$and$$(n+1)^2$$ ${(n+1)}^2$. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between$$n^2$$ $n^2$and$$n(n+1)$$ $n(n+1)$and also between$$n(n+1)$$ $n(n+1)$and$$(n+1)^2$$ ${(n+1)}^2$. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all$$n\le N$$ $n\le N$in time$$O( N \log N \log \log N)$$ $O(NlogNloglogN)$and space$$N^{O(1/\log \log N)}$$ $N^{O(1/loglogN)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous$$N = 2\cdot 10^{9}$$ $N=2·{10}^9$up to$$N = 7.05\cdot 10^{13} > 2^{46}$$ $N=7.05·{10}^{13}>2^{46}$, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors.