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A138465
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Non-optimus primes.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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