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.
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
KEYWORD
nonn
AUTHOR
Tom Mueller, Apr 16 2007
STATUS
approved