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Bertrand's Postulate for Carmichael Numbers


Bertrand's postulate for Carmichael numbers is the theorem that, for every delta>0 and all sufficiently large x (depending on delta), there are at least exp((lnx)/((lnlnx)^(2+delta))) Carmichael numbers between x and x+x/(lnx)^(1/(2+delta)) (Larsen 2023). In particular, there is a Carmichael number between x and 2x for every sufficiently large x.

Alford et al. (1994) posed the question after proving that there are infinitely many Carmichael numbers. Larsen's result answers it with the stronger short-interval bound above.


See also

Bertrand's Postulate, Carmichael Number

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References

Alford, W. R.; Granville, A.; and Pomerance, C. "There Are Infinitely Many Carmichael Numbers." Ann. Math. 139, 703-722, 1994.Larsen, D. "Bertrand's Postulate for Carmichael Numbers." Int. Math. Res. Not. 2023, 13072-13098, 2023. https://doi.org/10.1093/imrn/rnac203.Turing. "17-Year-Old Solves a Three Decade Old Conjecture." Aug. 26, 2026. https://www.youtube.com/watch?v=bykiWGYmPgc.

Cite this as:

Weisstein, Eric W. "Bertrand's Postulate for Carmichael Numbers." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BertrandsPostulateforCarmichaelNumbers.html

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