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

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: A191086 A058302 A133213 * A006598 A245623 A106892

Adjacent sequences:  A138462 A138463 A138464 * A138466 A138467 A138468

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 07 2009

EXTENSIONS

More terms from Charles R Greathouse IV, Sep 26 2012

STATUS

approved

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Last modified June 26 12:37 EDT 2022. Contains 354883 sequences. (Running on oeis4.)