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QUADRATIC FORMS, RECIPROCITY LAWS, and PRIMES of the FORM X2 + Ny2

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QUADRATIC FORMS, RECIPROCITY LAWS, and PRIMES of the FORM X2 + Ny2 QUADRATIC FORMS, RECIPROCITY LAWS, AND PRIMES OF THE FORM x2 + ny2 ALEKSANDER SKENDERI Abstract. This paper demonstrates necessary and sufficient conditions, for several values of n, for when a prime p can be written in the form x2 + ny2. In doing this, we provide an introduction to quadratic forms and elementary genus theory. Moreover, we show how cubic and biquadratic reciprocity can be used to tackle our problems in certain more complicated cases. The paper assumes a good knowledge of quadratic reciprocity, as well as basic abstract algebra, such as familiarity with rings, fields, integral domains, and ideals. Contents 1. Introduction 1 2. The Theory of Quadratic Forms 1 3. Elementary Genus Theory 9 4. Reciprocity Laws 12 4.1. Cubic Reciprocity 12 4.2. Biquadratic Reciprocity 19 Acknowledgments 23 References 23 1. Introduction The question of when a prime can be written in the form x2 + ny2, where x and y are integers, began with Fermat, who conjectured necessary and sufficient conditions for primes of the form x2 + y2, x2 + 2y2, and x2 + 3y2. Several questions immediately come to mind: can we formulate similar results for other values of n? For how many values of n do we have such conditions? We will tackle the first question, proving Fermat's conjectures, in addition to addressing more advanced cases. Lagrange, Legendre, and Gauss made progress on this problem by first de- veloping a theory of quadratic forms. Gauss and Eisenstein worked on cubic and biquadratic reciprocity, which we will use to prove results for the cases when n = 27 or 64, respectively. As we shall soon see, the mathematics they developed is very beautiful as well as a great source of mathematical inspiration. 2. The Theory of Quadratic Forms The theory of quadratic forms, when used in conjunction with quadratic reci- procity, is exceedingly useful in giving necessary and sufficient conditions for when a prime may be represented in the form x2 + ny2. Only the notions of equivalence, 1 2 ALEKSANDER SKENDERI the discriminant, and reduced form need to be developed in order to prove the following beautiful results, which we will address at the end of this section: (1) p = x2 + y2; if and only if p ≡ 1 (mod 4); (2) p = x2 + 2y2; if and only if p ≡ 1; 3 (mod 8); (3) p = x2 + 3y2; if and only if p = 3 or p ≡ 1 (mod 3); (4) p = x2 + 7y2; if and only if p = 7 or p ≡ 1; 9; 11; 15; 23; 25 (mod 28). This is already remarkable, as it not only fully resolves, but also extends upon the conjectures made by Fermat. Once we introduce the additional notion of genus, we will prove similar results for many other cases, such as n = 6 or 10. Before tackling our main question, we need to develop a theory of quadratic forms. Definition 2.1. An integral quadratic form is a homogeneous polynomial of degree 2 of the form f(x; y) = ax2 + bxy + cy2, where a; b; and c are integers. Definition 2.2. A form ax2 + bxy + cy2 is primitive if its coefficients a; b and c are relatively prime. That is, gcd(a; b; c) = 1. Definition 2.3. An integer m is represented by a form f(x; y) if the equation m = f(x; y) has an integer solution in x and y. Moreover, if x and y are relatively prime, then we say that m is properly represented by f(x; y). Thus, we are interested in the following question: when is a prime represented by the quadratic form x2 + ny2? Studying this question requires the notion of equivalence of quadratic forms. Definition 2.4. Two forms f(x; y) and g(x; y) are equivalent if there exist integers p; q; r and s such that ps − qr = ±1 and, f(x; y) = g(px + qy; rx + sy): If p; q; r and s can be chosen so that ps − qr = 1, then the equivalence is called proper. Notice that going from one form to an equivalent form amounts to nothing else than changing basis, and requiring that the determinant of the matrix p q M = r s be ±1. An important property of equivalent forms, that follows directly from the definition (and is what motivates the definition), is that equivalent forms represent the same numbers. Moreover, properly equivalent forms properly represent the same numbers. The following is a very useful lemma. QUADRATIC FORMS, RECIPROCITY LAWS, AND PRIMES OF THE FORM x2 + ny2 3 Lemma 2.5. A form f(x,y) properly represents an integer m if and only if f(x,y) is properly equivalent to the form mx2 + Bxy + Cy2 for some B; C 2 Z. Proof. First, suppose that f(p; q) = m, where gcd(p; q) = 1. By the Euclidean Algorithm, there exist integers r and s such that ps − qr = 1. By writing f(x; y) = ax2 + bxy + cy2, a short calculation shows that f(px + ry; qx + sy) = f(p; q)x2 + (2apr + bps + brq + 2cqs)xy + f(r; s)y2 = mx2 + Bxy + Cy2; where B = 2apr + bps + brq + 2cqs; and C = f(r; s): Conversely, suppose that there exist integers p; s; q and r with ps − qr = 1 such that f(px + ry; qx + sy) = mx2 + Bxy + Cy2; for some integers B and C. Notice that (x; y) = (1; 0) gives f(p; q) = m, where gcd(p; q) = 1, since otherwise we could not have ps − qr = 1. Definition 2.6. The discriminant D of a quadratic form f(x; y) = ax2 + bxy + cy2 is D = b2 − 4ac. Remark 2.7. Suppose that f(x; y) and g(x; y) are equivalent forms. That is, f(x; y) = g(px + qy; rx + sy), with discriminants D and D0, respectively. Then, a small calculation shows that (2.8) D = (ps − qr)2D0: Therefore, D = D0 since ps − qr = ±1. Thus, equivalent forms have the same discriminant. However, forms with the same discriminant are not always equivalent, as we shall see in Remark 2.12. Another important feature of the discriminant is that it has a strong effect on the sign of the values represented by the corresponding quadratic form. Let f(x; y) = ax2 + bxy + cy2 be a form of discriminant D. Then, another little calculation verifies that (2.9) 4af(x; y) = (2ax + by)2 − Dy2: We call a form positive definite if it only represents non-negative integers, and negative definite if it only represents non-positive integers. If D < 0, then the right-hand side is always non-negative. Thus, f(x; y) is positive definite if a > 0, and negative definite if a < 0. Lemma 2.10. Let D ≡ 0; 1 (mod 4) be an integer and m an odd integer relatively prime to D. Then m is properly represented by a primitive form of discriminant D if and only if D is a quadratic residue modulo m. Proof. If a primitive form f(x; y) of discriminant D properly represents m, then by Lemma 2.5, f(x; y) is properly equivalent to the form mx2 + bxy + cy2 for some integers b and c. By the above remark, these two forms have the same discriminant and so we see that D = b2 − 4mc ≡ b2 (mod m), as desired. Conversely suppose that D ≡ b2 (mod m). Since m is odd, we can assume D and b have the same parity. From the fact that D ≡ 0; 1 (mod 4), we see that D ≡ b2 (mod 4), and thus D ≡ b2 (mod 4m), because gcd(4; m) = 1. Therefore, 4 ALEKSANDER SKENDERI D = b2 − 4mc, for some c. Then, the form mx2 + bxy + cy2 properly represents m, has discriminant D, and has relatively prime coefficients, since m and D are relatively prime. The lemma gives the following very useful corollary. Corollary 2.11. Let n be an integer and p an odd prime not dividing n. Then −n = 1 if and only if p is properly represented by a primitive form of p discriminant −4n. Proof. By the properties of the Legendre symbol, we have −4n 22−n −n = = ; p p p p and so −4n is a quadratic residue modulo p if and only if −n is a quadratic residue modulo p. Applying Lemma 2.10 completes the proof. Remark 2.12. Notice that this corollary is relevant to our question of which primes are of the form x2 + ny2, since x2 + ny2 is primitive and has discriminant −4n. The problem is that there are many forms of the same discriminant. For example, x2 + 5y2 and 2x2 + 2xy + 3y2 are both forms of discriminant −20, but they are not equivalent. Both forms are positive definite by (2.9), but x2 + 5y2 represents 1 (let (x; y) = (1; 0)), whereas 2x2 + 2xy + 3y2 does not. In fact the smallest positive integer that 2x2 + 2xy + 3y2 represents is 2 (let (x; y) = (1; 0)). Therefore, we need to show that every quadratic form is equivalent to a very simple one. Definition 2.13. A primitive positive definite form ax2 + bxy + cy2 is said to be reduced if jbj ≤ a ≤ c and b ≥ 0 if either jbj = a or a = c. Primitive positive definite forms have a very important property, illustrated in the following theorem. Theorem 2.14. Every primitive positive definite form is properly equivalent to a unique reduced form. The proof is long, tedious, and not very illuminating. See Theorem 2.8 of Cox [1] for the details. Theorem 2.14 allows us to sometimes determine whether forms are merely equiv- alent, or actually properly equivalent.
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