voightalmost

voightalmost - arXiv:math/0410266v2[math.NT 16 Sep 2005...

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Unformatted text preview: arXiv:math/0410266v2 [math.NT] 16 Sep 2005 MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0 QUADRATIC FORMS THAT REPRESENT ALMOST THE SAME PRIMES JOHN VOIGHT Abstract. Jagy and Kaplansky exhibited a table of 68 pairs of positive def- inite binary quadratic forms that represent the same odd primes and conjec- tured that their list is complete outside of “trivial” pairs. In this article, we confirm their conjecture, and in fact find all pairs of such forms that represent the same primes outside of a finite set. 1. Introduction The forms x 2 + 9 y 2 and x 2 + 12 y 2 represent the same set of prime numbers, namely, those primes p which can be written p = 12 n + 1 for some positive integer n . What other like pairs of forms exist? Jagy and Kaplansky [JK] performed a computer search for pairs that represent the same set of odd primes and found cer- tain “trivial” pairs which occur infinitely often and listed other sporadic examples. They conjecture that their list is complete. Using the tools of class field theory, in this article we give a provably complete list of such pairs. By a form Q we mean an integral positive definite binary quadratic form Q = ax 2 + bxy + cy 2 ∈ Z [ x,y ]; the discriminant of Q is b 2 − 4 ac = D = df 2 < 0, where d is the discriminant of Q ( √ D ) or the fundamental discriminant , and f ≥ 1. We will often abbreviate Q = ( a,b,c ) . Throughout, we look for forms that represent the same primes outside of a finite set—we say then that they represent almost the same primes . A form represents the same primes as any equivalent form under the action of the group GL 2 ( Z ). Hence from now on (except in the statement of Proposition 3.4, see Remark 3.5, and in the proof of Lemma 3.15), we insist that a form be GL 2 ( Z )-reduced , i.e., 0 ≤ b ≤ a ≤ c . Moreover, the set of primes represented by a form is finite (up to a finite set, it is empty) if and only if the form is nonprimitive, that is to say gcd( a,b,c ) > 1, and any two nonprimitive forms represent almost the same primes. We therefore also insist that a form be primitive , so that the set of primes represented is infinite. If Q 1 ,Q 2 are forms which represent almost the same primes, we write Q 1 ∼ Q 2 ; it is clear that ∼ defines an equivalence relation on the set of forms. To every equivalence class C of forms, we associate the set δ ( C ) of fundamental discriminants d of the forms in C as well as the set Δ( C ) of discriminants D of forms in C . Received by the editor September 16, 2005. 1991 Mathematics Subject Classification. Primary 11E12; Secondary 11E16, 11R11. Key words and phrases. Binary quadratic forms, number theory....
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voightalmost - arXiv:math/0410266v2[math.NT 16 Sep 2005...

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