A002052 - OEIS

%I M4339 N1816 #27 Feb 04 2022 00:59:27

%S 3,7,11,19,23,31,43,47,59,67,71,83,103,107,127,131,139,151,163,167,

%T 179,191,199,211,227,239,251,263,271,283,307,311,331,347,367,379,383,

%U 419

%N Prime determinants of forms with class number 2.

%C 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

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Rick L. Shepherd, <a href="/A002052/b002052.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Suryanarayana, <a href="https://www.ias.ac.in/article/fulltext/seca/002/02/0178-0179">Positive determinants of binary quadratic forms whose class-number is 2</a>, Proceedings of the Indian Academy of Sciences. Section A, 2 (1935), 178-179.

%o (PARI) {QFBclassno(D) = qfbclassno(D) * if(D < 0 || norm(quadunit(D)) < 0, 1, 2);

%o 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

%Y Cf. A260335. Subsequence of A002145.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Term 3 added by _Rick L. Shepherd_, Apr 29 2015