A107662 - OEIS

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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.

23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907 (list; graph; refs; listen; history; internal format)

	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	`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	`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: A133659 A106312 A023679 * A083370 A124582 A130796` `Adjacent sequences: A107659 A107660 A107661 * A107663 A107664 A107665`
	KEYWORD	`fini,full,nonn`
	AUTHOR	`T. D. Noe (noe(AT)sspectra.com), May 19 2005`

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Last modified February 16 21:30 EST 2012. Contains 205971 sequences.