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A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form. 0

%I

%S 23,31,44,59,76,83,107,108,139,172,211,243,268,283,307,331,379,499,

%T 547,643,652,883,907

%N -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.

%C Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.

%D Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

%D Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ClassNumber.html">Class Number</a>

%e For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).

%Y Cf. A106312 (possible negative discriminants of cubic polynomials), A014602 (negative discriminants having class number 1), A006203 (negative discriminants having class number 3).

%K fini,full,nonn

%O 1,1

%A _T. D. Noe_, May 19 2005

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