A138465 Non-optimus primes.
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
Keywords
Examples
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.
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.
- Arkadii Slinko, Additive Representability of Finite Measurement Structures, 2007, 26 pp. [Cached copy]
Crossrefs
Cf. A217090 (optimus primes).
Programs
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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
Extensions
More terms from Charles R Greathouse IV, Sep 26 2012
Comments