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On Triangles As Real Analytic Varieties of an Extended Fermat Equation

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On Triangles As Real Analytic Varieties of an Extended Fermat Equation On Triangles as Real Analytic Varieties of an Extended Fermat Equation Giri Prabhakar∗ Abstract On the one hand, triangles play an important role in the solution of elliptic curves, as exemplified in studies by Tunnell on the congruent number problem, or Ono relating arbitrary triangles to elliptic curves. On the other hand, Frey-Hellegouarch curves are deeply connected to Fermat's equation. Thus, we are motivated to explore the relationship between triangles and Fermat's equation. We show that triangles in Euclidean space can be exactly described by real analytic varieties of an extension of the Fermat equation. To the 3 best of our knowledge, this formulation has not been explored previously. Let S = f(a; b; c) 2 R≥0 j (c > φ φ φ 0 ^ 9 φ 2 R≥1)[a + b = c ]g be the set of real non-negative solutions to the Fermat equation extended to admit real valued exponents φ ≥ 1, which we term the Fermat index. Let E be the set of triplets of side lengths of all non-degenerate and degenerate 2-simplices. We show that S = E, and that each triangle has a unique Fermat index. We show that there exists a deformation retraction F based on the extended Fermat equation, that maps all possible sides a ≤ c and b ≤ c of 2-simplices (a; b; c) 2 S, to the retract c of fixed length at a fixed angle to a. However, (a; b; c) is also governed by the law of cosines; thus an alternate map P exists based on the law of cosines, which must be geometrically identical to F . Setting F = P generates an algebraic equation that yields a class of polynomials closely related to the 2mth cyclotomic polynomial. We propose criteria for the irreducibility of these polynomials in Q[x], and present an extension of the Schoenemann-Eisenstein theorem, which we employ along with additive shifts and Gauss's Lemma to prove the propositions. The approach yields new geometric insights about triangles: in particular, a simple proof that no rational triangle exists with a Fermat index of 4, and in general, the result that no non-degenerate oblique rational triangle can have an integer Fermat index. Keywords: Euclidean geometry, Pythagorean theorem, Diophantine equations 2010 MSC: 51N20, 51M04, 51M05 1. Introduction Trigonometric methods are indispensable in a wide spectrum of modern scientific and technological fields, but the fundamental representations and analyses of triangles is one of the oldest and most established subjects [1]. Therefore, the expectation that studies in these areas have little that is new to offer should come as no surprise. Nevertheless, there is still a possibility that one comes across undiscovered insight, such as the work by Kendig extending the interpretation of Heron's formula to the complex plane[2]. Considerations in plane trigonometry result from the root presumption of the Pythagorean theorem. The cosine law is a prime example that relates all three sides of a triangle, and we will call this the Pythagorean representation of a triangle. Note that the term \triangle" in this paper refers to 2-simplices in Euclidean space. On the one hand, many studies arising from the congruent number problem and generalizations of this problem have shown that rational triangles are closely related to solutions of elliptic curves[1, 3{6]. Tunnell related the congruent number problem to the Birch and Swinnerton-Dyer conjecture [6]. Ono describes ∗Corresponding author Email address: [email protected] (Giri Prabhakar) On Triangles and an Extended Fermat Equation Giri Prabhakar a method by which it is possible to generate an infinite number of elliptic curves over an algebraic field by replacing right triangles in the congruent number problem with arbitrary triangles [7]. On the other hand, elliptic curves are also strongly connected to Fermat's equation; Frey-Hellegouarch curves of the form 2 m m m m m 3 y = x(x − a )(x + b ) are connected to Fermat's equation a + b = c where (a; b; c) 2 Z>0 and m is an odd prime [8]. The contributions of Frey, Serre, Shimura, Taniyama and others on the modularity of these elliptic curves over the field of rational numbers played a critical role in the breakthrough and celebrated proof of Fermat's theorem by Wiles[8{15]. The strong relationship between triangles and elliptic curves on the one hand, and elliptic curves and Fermat's equation on the other, motivated us to study the connection between triangles and an extension of Fermat's equation. We have discovered an alternate representation of the relationship between the sides of a triangle, that is not a general consequence of the Pythagorean theorem. We call this the Fermat representation. In addition to making connections to Kendig's observations and cyclotomic polynomials, we also present an extension of the Schoenemann-Eisenstein theorem and fresh insight into the properties of triangles that, to the best of our knowledge, have been previously unexplored. We begin in this section by showing that the extended Fermat equation offers an alternate, exact, real analytic description of any triangle. It is naturally interesting to then consider the relationship between the Fermat and Pythagorean representations, and its consequent implications. Therefore, in section 2, we estab- lish the geometric framework which is the basis upon which the Fermat and Pythagorean representations are related in section 3. The relationship suggests the formulation of an analytic function, which takes the form of univariate polynomials for positive integer values of the exponent. We then propose the irreducibility of the resultant polynomials in section 4, and prove these propositions by means of an extension of the Schoenemann-Eisenstein theorem [16] and Gauss's Lemma [17]; these are our main results. 3 φ φ φ Theorem 1. Let S = f(a; b; c) 2 R≥0 j (c > 0 ^ 9 φ 2 R≥1)[a + b = c ]g be triplets of non-negative real numbers that satisfy aφ + bφ = cφ: (1) The number φ will be called Fermat index. Let E be the set of triplets of side lengths of all non-degenerate and degenerate 2-simplices. Then, S = E, and each 2-simplex has a unique Fermat index. Proof. First let b > a and φ 2 (1; 1); and consider the expression (a + b)φ = bφ[1 + (a=b)]φ; φ which, on expanding [1 + (a=b)] as a convergent binomial series with k 2 Z≥0, becomes 1 X φ φ bφ (a=b)k, where := φ(φ − 1) ::: (φ − k + 1)=k!; k k k=0 1 X φ = bφ + φabφ−1 + bφ (a=b)k; k k=2 which holds for both integer and non-integer real values of φ. Since aφ−1 < φbφ−1, 1 X φ (a + b)φ > bφ + aφ + bφ (a=b)k: (2) k k=2 th P1 φ k The k term of k=2 k (a=b) is k tk = [φ(φ − 1) ::: (φ − k + 1)=k!](a=b) ; so that tk + tk+1 = tkf1 + [(φ − k)=(k + 1)](a=b)g: 2 On Triangles and an Extended Fermat Equation Giri Prabhakar n Let n be the first integer that is greater than φ; then tn = [φ(φ − 1) ::: (φ − n + 1)=n!](a=b) ≥ 0, and in fact φ 8 i 2 Z≥0 3 r = n + 2i; tr ≥ 0, because r is either 0 (when φ is an integer) or contains the product of an even number of negative terms in the numerator (when φ is a non-integer real number). Since 1 < φ < r, it follows that r − φ < r − 1 < r + 1. Therefore, φ − r < 0 and j(φ − r)=(r + 1)j < 1 =) 0 < tr + tr+1 ≤ tr, with the equality applying when φ = n − 1. Hence 1 1 n−1 X φ X X (a=b)k = t = t + (t + t ) + (t + t ) + :::; k k k n n+1 n+2 n+3 k=2 k=2 k=2 which implies that n−1 1 X X φ 0 < t < (a=b)k < (1 + a=b)n; k k k=2 k=2 and from (2) leading to the inequality (a + b)φ > aφ + bφ = cφ; from which we conclude that a + b > c. Since c > a and c > b, we get b + c > a and c + a > b. Thus, for a 6= b and φ > 1, (1) implies the triangle inequalities, which are necessary and sufficient for the triplet (a; b; c) to form a 2-simplex [18]. When φ = 1, (1) implies that (a; b; c) is a degenerate 2-simplex. For a = b, the triangle inequality is trivially satisfied in S, since in (1), 21/φa = c (taking only the positive real root), thus 2a > c, but also c > a. Clearly, a triplet (a; b; c) satisfying (1) is equivalent to (λa, λb, λc) 8 λ 2 R, and hence S ⊂ P2(R). Conversely, consider a triangle (a; b; c) 2 E which is either degenerate or non-degenerate (in the former case with c and at least one of a, b non-zero), with the length of all non-zero sides greater than 1. The assumption about the length does not lose generality, since E ⊂ P2(R), but is simpler for analysis. Let c be (one of) the longest side(s). We have, in the degenerate case (without loss of generality) a+b = c; a < c; b ≤ c, and in the non-degenerate case, a + b > c, a ≤ c, b ≤ c. First we assume strict inequality, and defer the equality cases to Lemmas 2 and 3 (where they will be shown to correspond to the extreme cases φ ! 1 and φ ! 1).
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