On the distribution of pseudoprimes
HTML articles powered by AMS MathViewer
- by Carl Pomerance PDF
- Math. Comp. 37 (1981), 587-593 Request permission
Abstract:
Let $\mathcal {P}(x)$ denote the pseudoprime counting function. With \[ L(x) = \exp \{ \log x\log \log \log x/\log \log x\} ,\] we prove $\mathcal {P}(x) \leqslant x \bullet L{(x)^{ - 1/2}}$ for large x, an improvement on the 1956 work of Erdös. We conjecture that $\mathcal {P}(x) = x \bullet L{(x)^{ - 1 + o(1)}}$.References
-
N. G. de Bruijn, "On the number of positive integers $\leqslant x$ and free of prime factors $> y$. II," Nederl. Akad. Wetensch. Proc. Ser. A, v. 69, 1966, pp. 239-247.
- E. Rodney Canfield and Carl Pomerance, On the problem of uniqueness for the maximum Stirling number(s) of the second kind, Integers 2 (2002), A1, 13. MR 1876176
- P. Erdös, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), 201–206. MR 79031
- P. Erdős, On the sum $\sum _{d2^{n}-1}d^{-1}$, Israel J. Math. 9 (1971), 43–48. MR 269613, DOI 10.1007/BF02771618
- Carl Pomerance, Popular values of Euler’s function, Mathematika 27 (1980), no. 1, 84–89. MR 581999, DOI 10.1112/S0025579300009967
- Carl Pomerance, A new lower bound for the pseudoprime counting function, Illinois J. Math. 26 (1982), no. 1, 4–9. MR 638549
- Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff Jr., The pseudoprimes to $25\cdot 10^{9}$, Math. Comp. 35 (1980), no. 151, 1003–1026. MR 572872, DOI 10.1090/S0025-5718-1980-0572872-7 R. A. Rankin, "The difference between consecutive prime numbers," J. London Math. Soc., v. 13, 1938, pp. 242-247.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 37 (1981), 587-593
- MSC: Primary 10A21; Secondary 10A20
- DOI: https://doi.org/10.1090/S0025-5718-1981-0628717-0
- MathSciNet review: 628717