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%I #21 May 04 2017 11:35:36
%S 3,23,31,137,191,239,277,359,431,439,683,719,743,911,997,1031,1061,
%T 1103,1109,1223,1279,1423,1439,1481,1511,1559,1583,1597,1733,1873,
%U 2017,2039,2063,2351,2399,2411,2543,2683,2897,2903,3023,3347,3359,3457,3517,3607,3623,3793,3797
%N Non-optimus primes.
%C 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.
%D 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.
%H Charles R Greathouse IV, <a href="/A138465/b138465.txt">Table of n, a(n) for n = 1..5000</a>
%H S. Marshall, <a href="http://www.sipta.org/isipta05/proceedings/papers/s030.pdf">On the existence of extremal cones and comparative probability orderings</a>, Proceedings of The 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA 05), Pittsburg, Pennsylvania, 2005, pp. 246-255.
%H Arkadii Slinko, <a href="http://www.math.auckland.ac.nz/~slinko/Research/survey5.pdf">Additive Representability of Finite Measurement Structures</a>, 2007, 26 pp.
%H Arkadii Slinko, <a href="/A217090/a217090.pdf">Additive Representability of Finite Measurement Structures</a>, 2007, 26 pp. [Cached copy]
%e 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.
%e 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.
%o (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
%Y Cf. A217090 (optimus primes).
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Feb 07 2009
%E More terms from _Charles R Greathouse IV_, Sep 26 2012
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