Elite prime number

In number theory , a prime number is called elitist if only finitely many Fermat numbers are quadratic remainders modulo . ${\displaystyle p}$ ${\displaystyle F_{n}=2^{2^{n}}+1}$ ${\displaystyle p}$

They owe their name to the Austrian mathematician Alexander Aigner, who described them in 1986 and was the first to study them. Aigner called these prime numbers elitist because they only appear very rarely; he himself only found 14 such prime numbers that are smaller than 35,000,000.

Since Fermat numbers satisfy the relationship , the congruence sequence ( mod ) becomes periodic from a certain index , i.e. H. there is a minimal natural number such that (mod ) holds for all natural numbers . The terms are called Fermat residues of . Accordingly, a prime number is elitist if and only if all Fermat residues are quadratic non-residues modulo . ${\displaystyle F_{n+1}=(F_{n}-1)^{2}+1}$ ${\displaystyle F_{n}}$ ${\displaystyle p}$ ${\displaystyle s}$ ${\displaystyle L}$ ${\displaystyle F_{s+k+L}\equiv F_{s+k}}$ ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle F_{s},F_{s+1},\ldots ,F_{s+L-1}}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

The first elite prime numbers are: 3, 5, 7, 41, 15,361, 23,041, 26,881, 61,441, 87,041, 163,841, ... (sequence A102742 in OEIS )

It is unknown whether there are an infinite number of elite prime numbers. However, it could be shown that the number of all elite prime numbers fulfills the estimate . ${\displaystyle C(x)}$ ${\displaystyle \leq x}$ ${\displaystyle C(x)=O\left({\tfrac {x}{\ln ^{2}(x)}}\right)}$

Individual evidence

1. ↑ A. Aigner: About prime numbers, according to which (almost) all Fermat numbers are quadratic non-residues. In: Monthly Mathematics. 101, 1986, pp. 85-93.
2. ^ Krizek et al .: On the convergence of series of reciprocals of prims related to the Fermat numbers. In: Journal of Number Theory. 97, 2002, pp. 95-112.

Web links

• Alain Chaumont, Tom Müller: All Elite Primes Up to 250 Billion . In: Journal of Integer Sequences . tape 9 , no. 06.3.8 , 2006 ( cs.uwaterloo.ca [PDF]).