

A002052


Prime determinants of forms with class number 2.
(Formerly M4339 N1816)


3



3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 367, 379, 383, 419
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OFFSET

1,1


COMMENTS

The Suryanarayana paper contains these errors: In section 2, list (1) omits 3 and an asterisk should follow 1987; list (2) should include neither 3203 nor 3271. Section 3 should say "Of the 339 primes d == 3 (4) up to 5000, 289 primes satisfy h(d) = 2, while 50 do not." (correcting all three counts)  Rick L. Shepherd, Apr 29 2015
Also primes p > 2 such that Z[sqrt(p)] = Z[x]/(x^2  p) is a unique factorization domain (or equivalently, a principal ideal domain). This can be deduced from the following result: let K be the quadratic field with discriminant D > 0, h(D) and h_+(D) be the ordinary class number and narrow class numer (or form class number) of K respectively, then h_+(D)/h(D) = 1 if the fundamental unit of K has norm 1; 2 if the fundamental unit of K has norm 1.  Jianing Song, Feb 17 2021


REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
M. Suryanarayana, Positive determinants of binary quadratic forms whose classnumber is 2, Proceedings of the Indian Academy of Sciences. Section A, 2 (1935), 178179.


PROG

(PARI) {QFBclassno(D) = qfbclassno(D) * if(D < 0  norm(quadunit(D)) < 0, 1, 2);
n=0; forprime(p=3, 291619, if(p%4 == 3 && QFBclassno(4*p) == 2, n++; write("b002052.txt", n, " ", p)))} \\ Rick L. Shepherd, Apr 29 2015


CROSSREFS

Cf. A260335. Subsequence of A002145.
Sequence in context: A181516 A285015 A002145 * A092109 A117991 A118260
Adjacent sequences: A002049 A002050 A002051 * A002053 A002054 A002055


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Term 3 added by Rick L. Shepherd, Apr 29 2015


STATUS

approved



