A102742 Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.

3, 5, 7, 41, 15361, 23041, 26881, 61441, 87041, 163841, 544001, 604801, 6684673, 14172161, 159318017, 446960641, 1151139841, 3208642561, 38126223361, 108905103361, 171727482881, 318093312001, 443069456129, 912680550401, 1295536619521, 1825696645121, 2061584302081

Offset: 1

Author

Tom Mueller, Feb 08 2005; extended Jun 16 2005

Keywords

• nonn

Comments

Křížek, Luca, Shparlinski, & Somer show that a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 25 2017

Let d = 2^r*d' be the multiplicative order of 2 modulo p. Note that 2^2^s == 2^d == 1 (mod p), so p divides none of.

Links

• Dennis Martin, Table of n, a(n) for n = 1..29
• Alexander Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind, Monatsh. Math., Vol. 101 (1986), pp. 85-93; alternative link.
• Alain Chaumont and Tom Mueller, All Elite Primes Up to 250 Billion, J. Integer Sequences, Vol. 9 (2006), Article 06.3.8.
• Matthew Just, On upper bounds for the count of elite primes, arXiv:2102.00906 [math.NT], 2021.
• Michal Křížek, Florian Luca, Igor E. Shparlinski, and Lawrence Somer, On the complexity of testing elite primes, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.2, 5 pp.
• Xiaoquin Li, Verifying Two Conjectures on Generalized Elite Primes, JIS 12 (2009) 09.4.7.
• Dennis Martin, Elite Prime Search. [Broken link]
• Dennis Martin, Elite Prime Search. [Cached copy, with permission of author]
• Dennis Martin, Elite and Anti-Elite Prime Search Methodology [Cached copy, with permission of author]
• Tom Müller, Searching for large elite primes, Experimental Mathematics, Vol. 15, Nol. 2 (2006), pp. 183-186.
• Tom Muller and A. Reinhart, On generalized Elite Primes, Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.1.
• Tom Müller, On the Fermat Periods of Natural Numbers, Journal of Integer Sequences, Vol. 13 (2010), Article 10.9.5.
• Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Crossrefs

Cf. A128852, A344785. Subsequence of A129802.

Programs

PARI

list_upto(N)={forprime(p=3,N,r=2^valuation(p-1,2); a=Mod(3,p); v=List(); k=0; while(1,listput(v,a); a=(a-1)^2+1; for(j=1,#v,if(v[j]==a,k=j;break(2)))); for(i=k,#v,znorder(v[i]) % r != 0 && next(2)); print1(p,", "))} \\ Slow, only for illustration, Jeppe Stig Nielsen, Jan 28 2020

PARI

isElite(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2,n)), StartPoint = valuation(d,2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(issquare(Mod(2,n)^2^i + 1), return(0))); 1, 0) \\ Jianing Song, May 15 2024

Formula

Sum_{n>=1} 1/a(n) = A344785. - Amiram Eldar, May 30 2021

Extensions

a(17) from Tom Mueller, Jul 20 2005

a(18)-a(21) from Tom Mueller, Apr 18 2006

6 further terms from Tom Mueller, Apr 16 2007