arXiv:2111.06963math.NT
Bertrand's Postulate for Carmichael Numbers
Abstract
Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all $δ>0$ and $x$ sufficiently large in terms of $δ$, there exist at least $e^{\frac{\log x}{(\log\log x)^{2+δ}}}$ Carmichael numbers between $x$ and $x+\frac{x}{(\log x)^{\frac{1}{2+δ}}}$.
Summary: 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3. 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561…
renewiltordView on HNParent context
Good tip. I should have just linked the arxiv.org page for the article and not the PDF directly https://arxiv.org/abs/2111.06963
renewiltordView on HNParent context
Interesting. Entering [ https://ar5iv.org/abs/2111.06963 ] also gives me the HTML render of the paper. Saves one step, since I don't have to fetch the PDF first. :-)
sohkamyungView on HNParent context