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A138465
Non-optimus primes.
2
3, 23, 31, 137, 191, 239, 277, 359, 431, 439, 683, 719, 743, 911, 997, 1031, 1061, 1103, 1109, 1223, 1279, 1423, 1439, 1481, 1511, 1559, 1583, 1597, 1733, 1873, 2017, 2039, 2063, 2351, 2399, 2411, 2543, 2683, 2897, 2903, 3023, 3347, 3359, 3457, 3517, 3607, 3623, 3793, 3797
OFFSET
1,1
COMMENTS
A prime p is an optimus prime if (1 + sqrt( legendre(-1,p)*p ))^p - 1 = r + s*sqrt( legendre(-1,p)*p ) where gcd(r,s) = p.
REFERENCES
A. Slinko, Additive representability of finite measurement structures, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 113-133.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..5000
S. Marshall, On the existence of extremal cones and comparative probability orderings, Proceedings of The 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA 05), Pittsburg, Pennsylvania, 2005, pp. 246-255.
Arkadii Slinko, Additive Representability of Finite Measurement Structures, 2007, 26 pp. [Cached copy]
EXAMPLE
For p = 13, (1 + sqrt( legendre(-1,p)*p ))^p - 1 = 209588223+58200064*13^(1/2), and gcd(209588223,58200064) = 13, so 13 is an optimus prime.
For p = 23, (1 + sqrt( legendre(-1,p)*p ))^p - 1 = 7453766387236863-24397683359744*(-23)^(1/2), but gcd(7453766387236863,24397683359744) = 1081 != 23, so 23 is a non-optimus prime.
PROG
(PARI) is(p)=if(p<3 || !isprime(p), return(0)); my(t=(2*quadgen(kronecker(-1, p)*p))^p); gcd(imag(t), real(t)-1)!=p \\ Charles R Greathouse IV, Sep 26 2012
CROSSREFS
Cf. A217090 (optimus primes).
Sequence in context: A058302 A133213 A368691 * A006598 A245623 A106892
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 07 2009
EXTENSIONS
More terms from Charles R Greathouse IV, Sep 26 2012
STATUS
approved