The relation between a generalized Fibonacci sequence and the length of Cunningham chains

2022/05/16 by Kanado, Yuya · 1 citation
#11A41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.07650

Abstract

Let p be a prime number. A chain \p,2p+1,4p+3,⋯,(p+1)2^l(p)-1-1\ is called the Cunningham chain generated by p if all elements are prime number and (p+1)2^l(p)-1 is composite. Then l(p) is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime p≤ N such that l(p)≥ k is asymptotically equal to B_k N/(log N)^k with a real B_k>0 for all natural number k. This suggests that l(p)=Ω(log p/loglog p). However, so far no good estimation is known. It has not even been proven whether \limsup_p→∞ l(p) is infinite or not. All we know is that l(p)=5 if p=2 and l(p)

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