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A Look at the ABC Conjecture Via Elliptic Curves

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A Look at the ABC Conjecture Via Elliptic Curves A Look at the ABC Conjecture via Elliptic Curves Nicole Cleary Brittany DiPietro Alexander Hill Gerard D.Koffi Beihua Yan Abstract We study the connection between elliptic curves and ABC triples. Two important results are proved. The first gives a method for finding new ABC triples. The second result states conditions under which the power of the new ABC triple increases or decreases. Finally, we present two algorithms stemming from these two results. 1 Introduction The ABC conjecture is a central open problem in number theory. It was formu- lated in 1985 by Joseph Oesterl´eand David Masser, who worked separately but eventually proposed equivalent conjectures. Like many other problems in number theory, the ABC conjecture can be stated in relatively simple, understandable terms. However, there are several profound implications of the ABC conjecture. Fermat's Last Theorem is one such implication which we will explore in the second section of this paper. 1.1 Statement of The ABC Conjecture Before stating the ABC conjecture, we define a few terms that are used frequently throughout this paper. Definition 1.1.1 The radical of a positive integer n, denoted rad(n), is defined as the product of the distinct prime factors of n. That is, rad(n) = Q p where p is prime. pjn 1 Example 1.1.2 Let n = 72 = 23 · 32: Then rad(n) = rad(72) = rad(23 · 32) = 2 · 3 = 6: Definition 1.1.3 Let A; B; C 2 Z. A triple (A; B; C) is called an ABC triple if A + B = C and gcd(A; B; C) = 1. Note: We can rearrange any given ABC triple so that A; B; C > 0. Hence, we assume in this paper that the ABC triples are positive. Example 1.1.4 By taking the absolute value of each element of (3; −7; 4), we can rearrange it as (3; 4; 7) to be an ABC triple. Remark 1.1.5 Since A + B = C, exactly one of the A; B or C must be divisible by 2. Hence, The product ABC is divisible by 2. log(C) Definition 1.1.6 Let (A; B; C) be an ABC triple. Then P (A; B; C) = log(rad(ABC)) is called the power. Example 1.1.7 If A = 3;B = 4;C = 7, then rad(3 · 22 · 7) = 3 · 2 · 7 = 42 and log(7) P (3; 4; 7) = log(42) ≈ 0:520: Note that most of the time, P (A; B; C) is less than 1. For our research, we focus on the ABC triples with power greater than 1. Example 1.1.8 If A = 1;B = 8, and C = A + B = 9, then rad(1 · 8 · 9) = 3 2 log(9) rad(8 · 9) = rad(2 · 3 ) = 2 · 3 = 6 and P (1; 8; 9) = log(6) ≈ 1:226. ABC triples with P (A; B; C) > 1 are considered exceptional . The highest power found thus far is 1:6299 and it corresponds to the ABC triple A = 2;B = 310 ·109; and C = 235: This triple was found by Eric Reyssat in 1987. Joseph Oesterl´e,one of the men that posed the ABC conjecture, was motivated by the Szpiro Conjecture, which encompasses many ideas of elliptic curves. The following is the first version of the ABC conjecture posed by Oesterl´e. Conjecture 1.1.9 (Oesterl´e,[8])For any " > 0, there exists only finitely many non-trivial ABC triples such that P (A; B; C) > 1 + ". A little later, David Masser developed the ABC conjecture while working with Mason's Theorem, which examines the degree of a polynomial and the radical of a polynomial. David Masser proposed this conjecture by taking Mason's Theorem and converting the polynomials into integers. The following is the equivalent version of the ABC conjecture presented by Masser. 2 Conjecture 1.1.10 (Masser,[8]) For every " > 0, there exist only finitely many non-trivial ABC triples such that C ≤ C(") · rad(ABC)1+", where C(") is a constant that depends only on ": The ABC conjecture proves to be one that requires more attention than is available during this seven week research program. As a result, we decide to use our resources to further explore the conjecture. In this paper, we focus on Oesterl´e's version of the ABC conjecture and its connection with elliptic curves. Through our examination of elliptic curves, we devise a method that leads to the discovery of ABC triples of powers 1:3428 and 1:4567. We also discover new ABC triples with significantly high merits of 23:710 and 23:68. We believe that with further research, additional high powered triples can be constructed using our methods. 2 Fermat's Last Theorem and the ABC Conjecture 2.1 Fermat's Last Theorem Fermat's Last Theorem was first posed in 1637 by Pierre de Fermat. By looking at the method used to prove Fermat's Last Theorem in 1994, we hope to gain insight to the ABC conjecture. We now explore a brief overview of the proof of the theorem to help demonstrate why an elliptic curve approach may yield results in the ABC conjecture. + n n Theorem 2.1.1 (Fermat's Last Theorem) For a; b; c 2 Z , the equation a +b = cn has no nonzero solutions for any integer n > 2. Fermat did not provide a proof for all n > 2, but he did prove the case when n = 4 and Euler was able to reduce to the case where n is an odd prime. After over 300 years, there was still no complete proof of Fermat's Last Theorem. In 1984, Gerhard Frey proposed another way of looking at the problem using the theory of elliptic curves. Definition 2.1.2 An elliptic curve E is a geometric object which can be mod- eled by an equation of the form E : y2 = x3 +Ax+B, where A and B are rational numbers such that ∆ = −16(4A3 + 27B2) 6= 0. Remark 2.1.3 ∆ = −16(4A3 + 27B2) 6= 0 is called the discriminant of the equation E : y2 = x3 + Ax + B. Gerhard Frey looked at an elliptic curve of the form E : y2 = x(x − ap)(x + bp), where p is an odd prime. Note that a and b in this equation correspond to Fermat's equation an + bn = cn. Gerhard Frey believed that if Fermat's equation has a 3 solution for p ≥ 5, then this curve would have such strange properties that it cannot exist. Frey's curve led to the proof of Fermat's Last Theorem in 1994 by Andrew Wiles. 2.2 Asymptotic Fermat's Last Theorem The motivation for our main approach to the ABC conjecture comes mostly from the proof of Fermat's Last Theorem. As we demonstrate, the ABC conjecture implies the asymptotic case of Fermat's Last Theorem. Though Fermat's Last Theorem took over 300 years to prove, the asymptotic case can be deduced by assuming the ABC conjecture. The asymptotic case of Fermat's Last Theorem states that there are only finitely many solutions to the equation an + bn = cn for n > 3. Proposition 2.2.1 The ABC conjecture implies the asymptotic case of Fermat's Last Theorem. n n n + Proof. Consider the solutions to a + b = c for n > 3 and a; b; c 2 Z with gcd(a; b; c) = 1. We investigate the power of these solutions: log(cn) P (an; bn; cn) = log(rad(anbncn)) n log(c) = log(rad(abc)) n log(c) ≥ log(rad(a) · rad(b) · rad(c)) n log(c) > log(c3) n log(c) n > = : 3 log(c) 3 n n n n Let A = a ;B = b and C = c . We have that P (A; B; C) > 3 > 1 + ". Now, by assuming the ABC conjecture, which states that there are only finitely many ABC triples with P (A; B; C) > 1 + ", we can then conclude that there are only n n n finitely many n for which Fermat's equation a + b = c holds true. By examining the proof of Fermat's Last Theorem and the asymptotic case of Fermat's Last Theorem, we develop some ideas on how to approach the ABC conjecture. It seems that examination through elliptic curves may be the most rewarding approach, and so we take a computational approach via elliptic curves. 4 3 Elliptic Curves There are many advantages in looking at elliptic curves. The algebraic group structure of elliptic curves can be used to implement new algorithms to find new ABC triples. Below, we state a few theorems about the group structure of elliptic curves. For more details on elliptic curves see [6]. 3.1 Group Structure of Elliptic Curves Recall that an elliptic curve E is a geometric object which can be modeled by an equation of the form E : y2 = x3 + Ax + B, where A and B are rational numbers such that ∆ = −16(4A3 + 27B2) 6= 0. Let E(Q) be the set of rational points on an elliptic curve E with the point at in- finity. We define a binary operation ⊕ on E(Q) making the ordered pair (E(Q); ⊕) an abelian group. In 1923, Mordell proved that E(Q) is finitely generated. Theorem 3.1.1 (Mordell,[5]) If E is an elliptic curve over Q, then the abelian group E(Q) is finitely generated. Furthermore, there exists a finite group E(Q)tors r and a nonnegative integer r such that E(Q) ' E(Q)tors × Z . The set E(Q)tors is the collection of points with finite order; it is called the torsion subgroup of E.
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