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A102742
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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Křížek, Luca, Shparlinski, & Somer show that a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 25 2017
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REFERENCES
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Alexander Aigner; Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), pp. 85-93
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LINKS
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Dennis Martin, Table of n, a(n) for n = 1..29
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 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
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 15:2 (2006), 183-186.
Tom Muller and A. Reinhart, On generalized Elite Primes, JIS 11 (2008) 08.3.1.
Tom Müller, On the Fermat Periods of Natural Numbers, J. Int. Seq. 13 (2010) # 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.
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PROG
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(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
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CROSSREFS
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Cf. A128852.
Sequence in context: A130536 A261511 A146972 * A089044 A117646 A064857
Adjacent sequences: A102739 A102740 A102741 * A102743 A102744 A102745
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KEYWORD
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nonn
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AUTHOR
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Tom Mueller, Feb 08 2005; extended Jun 16 2005
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EXTENSIONS
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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
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STATUS
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approved
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