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

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907

Offset

1,1

Comments

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.

References

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.

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.

Links

Table of n, a(n) for n=1..23.

Eric Weisstein's World of Mathematics, Class Number

Example

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).

Crossrefs

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

Sequence in context: A023679 A187773 A328825 * A256872 A276435 A083370

Adjacent sequences: A107659 A107660 A107661 * A107663 A107664 A107665

Keyword

fini,full,nonn

Author

T. D. Noe, May 19 2005

Status

approved