Nov 'ak-Carmichael numbers and shifted primes without large prime\n factors

2017/06/22 by Kalmynin, Alexander, Alexander Kalmynin
Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #History and Theory of Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1706.07343

openalex publication_date 2017/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove some new lower bounds for the counting function mathcal\nN _{mathcal C}(x) of the set of Nov 'ak-Carmichael numbers. Our estimates\ndepend on the bounds for the number of shifted primes without large prime\nfactors. In particular, we prove that mathcal N _{mathcal C}(x) \≫\nx^0.7039-o(1) unconditionally and that mathcal N _{mathcal C}(x) \≫\nxe^{-(7+o(1))(\log x)\(\log\log\log x)/(\log\log x)}, under some reasonable\nhypothesis.\n

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• On Novák numbers
• Carmichael numbers and least common multiples of p-1
• The Carmichael numbers up to 10¹⁸
• Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors
• The Carmichael numbers up to 10¹⁷