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A128852
Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.
7
2, 13, 17, 97, 193, 241, 257, 641, 673, 769, 2689, 5953, 8929, 12289, 40961, 49921, 61681, 65537, 101377, 114689, 274177, 286721, 319489, 414721, 417793, 550801, 786433, 974849, 1130641, 1376257, 1489153, 1810433, 2424833, 3602561, 6700417
OFFSET
1,1
COMMENTS
There are infinitely many anti-elite primes.
REFERENCES
Alexander Aigner; Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), pp. 85-93
LINKS
M. Krizek, F. Luca, I. E. Shparlinski, L. Somer, On the complexity of testing elite primes, J. Int. Seq. 14 (2011) # 11.1.2
Dennis Martin, Anti-Elite Prime Search [Cached copy, with permission of author]
Tom Müller, On Anti-Elite Prime Numbers, J. Integer Sequences, Vol. 10 (2007), Article 07.9.4.
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.
EXAMPLE
Let F_r:=2^(2^r)+1 = r-th Fermat number. Then a(2)=13 because for all r>1 we have F_r == 4 (mod 13) if r is even, resp. F_r == 10 (mod 13) if r is odd. Notice that 4 and 10 are quadratic residues modulo 13.
PROG
(PARI) isAntiElite(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, n == 2) \\ Jianing Song, May 15 2024
CROSSREFS
Cf. A102742.
Contains all Fermat prime factors of Fermat numbers (A023394) that are greater than 5.
Sequence in context: A109181 A175448 A067522 * A191765 A063615 A297837
KEYWORD
nonn
AUTHOR
Tom Mueller, Apr 16 2007
STATUS
approved