Integers of the form $$ax^2+bxy+cy^2$$ a x 2 + b x y + c y 2
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Integers of the form ax 2 + bxy + cy 2 Naoki Uchida1
Received: 4 March 2020 / Accepted: 5 August 2020 © Springer Nature Switzerland AG 2020
Abstract We provide a characterization of integers represented by the positive definite binary that D = b2 −4ac and quadratic form ax 2 +bx y +cy 2 . Suppose√ √ d K is the discriminant of the imaginary quadratic field K = Q( D). We call f = D/d K the conductor of ax 2 + bx y + cy 2 . In order to prove the main results, we define the “relative conductor” of two orders in an imaginary quadratic field. We provide a characterization of decomposition of proper ideals of orders in imaginary quadratic fields. Next, we provide characterizations of prime powers l h , where l divides the conductor, represented by the positive definite binary quadratic form ax 2 +bx y +cy 2 . Some interesting applications of the main results are also presented. For example, we provide an equivalent condition for when the equation m = 4x 2 + 2x y + 7y 2 has an integer solution. Note that its discriminant and conductor are − 108 and 6 and we do not assume that m is prime to 2 or 3. Keywords Binary quadratic forms · Representation of integers by quadratic forms · Diophantine equations · Orders in quadratic fields · Decomposition of ideals Mathematics Subject Classification 11E16 · 11E25 · 11N32 · 11R65
1 Introduction Fermat stated that for a prime p, the following equivalence holds: p = x 2 + y 2 has an integer solution ⇐⇒ p = 2 or p ≡ 1 mod 4. Many mathematicians, including Lagrange, Legendre and Gauss, developed genus theory and the theory of composition of quadratic forms. These theories enable us to
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Naoki Uchida [email protected] Tokyo University of Science, Noda Campus, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan
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prove that for a prime p, the following equivalence holds: p = 3x 2 + 2x y + 3y 2 has an integer solution ⇐⇒ p ≡ 3 mod 8. Gauss’ works introduce the notion of cubic reciprocity [4]. The following equivalence holds, for a prime p: p = 4x 2 + 2x y + 7y 2 has an integer solution ⇐⇒ p ≡ 1 mod 3 and 2 is not a cubic residue modulo p. Cox provided a characterization of primes of the form x 2 + ny 2, where n is a fixed positive integer [3]. It is natural to consider what happens if we replace a prime p with an arbitrary integer. There are some previous studies of this problem. For example, Fermat knew an equivalent condition for when integers can be written as x 2 + y 2, where x, y are some integers [5, Chapter 2]. Koo and Shin gave an equivalent condition for when the equation pq = x 2 + ny 2 has an integer solution, where n is a fixed positive integer, and p, q are distinct odd primes not dividing n [7]. Cho presented a characterization of integers, relatively prime to 2nm, represented by the form x 2 + ny 2 with x ≡ 1 mod m, y ≡ 0 mod m, where m, n are fixed positive integers [1]. He also presented a characterization of integers, relatively prime to 2(1 − 4n)m, represented by the form x 2 + x y + ny 2 with x ≡ 1 mod m, y ≡ 0 mod m, where m, n are fixed pos
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