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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

Dennis Martin, Table of n, a(n) for n = 1..101

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

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.

CROSSREFS

Cf. A102742.

Sequence in context: A109181 A175448 A067522 * A191765 A063615 A297837

Adjacent sequences:  A128849 A128850 A128851 * A128853 A128854 A128855

KEYWORD

nonn

AUTHOR

Tom Mueller, Apr 16 2007

STATUS

approved

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Last modified July 11 22:52 EDT 2020. Contains 335652 sequences. (Running on oeis4.)