THE EXPONENTIAL DIOPHANTINE EQUATION AX2 + BY 2 = λkZ AND ITS APPLICATIONS

Cao, Zhenfu; Chu, Chuan I; Shiu, Wai Chee

doi:10.11650/twjm/1500574244

Cited by 6 publications

(13 citation statements)

References 18 publications

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“…From Theorem 1, along with Lemma 1 from Section 2, we derive the following corollary which improves a result in [4] in which the bound depends on ω. Corollary 1. Let r and s be positive integers, let a and b be integers greater than 1, and let c be any odd positive integer prime to ra.…”

Section: Introductionmentioning

confidence: 61%

“…Also, note that (8) of Theorem 1 of [13] does not include the case (d 1 , d For this example we have (a, b, c, q, r, s, t) = (5, 11, 56, 1, 1, 5, 2). Other such examples are given by (a, b, c, q, r, s, t) = (2, 1, 3, 1, 1, 3, 2), (3, 5, 2, 1, 4, 3, 5), (5,3,2,1,4,3,7).…”

Section: Sharper Results For N ≤mentioning

confidence: 99%

“…Treatments using strictly elementary methods take advantage of the fact that c z is a perfect power, and thus are often sharper than those obtained by more general non-elementary methods, especially when c is divisible by few primes. These elementary bounds are dependent not only on n, where n is the number of distinct primes dividing XY , but also on ω, where ω is the number of distinct primes dividing c. Examples of such results are found in [9], [3], and [4]. In this paper we show that strictly elementary methods can be used to obtain a bound which is independent of ω (note that the bound of 2 16n+16 derived from the nonelementary result of Beukers and Schlickewei [1] is also independent of ω).…”

Section: Introductionmentioning

confidence: 98%

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Two terms with known prime divisors adding to a power

Scott¹,

Styer²

2018

Publ. Math. Debrecen

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Let c be a positive odd integer and R a set of n primes coprime with c. We consider equations X + Y = c z in three integer unknowns X, Y , z, where z > 0, Y > X > 0, and the primes dividing XY are precisely those in R. We consider N , the number of solutions of such an equation. Given a solution (X, Y, z), let D be the least positive integer such that (XY /D) 1/2 is an integer. Further, let ω be the number of distinct primes dividing c. Standard elementary approaches use an upper bound of 2 n for the number of possible D, and an upper bound of 2 ω−1 for the number of ideal factorizations of c in the field Q( √ −D) which can correspond (in a standard designated way) to a solution in which (XY /D) 1/2 ∈ Z, and obtain N ≤ 2 n+ω−1 . Here we improve this by finding an inverse proportionality relationship between a bound on the number of D which can occur in solutions and a bound (independent of D) on the number of ideal factorizations of c which can correspond to solutions for a given D. We obtain N ≤ 2 n−1 + 1. The bound is precise for n < 4: there are several cases with exactly 2 n−1 + 1 solutions.where c is a fixed positive odd integer, gcd(XY, c) = 1, and the set of primes in the factorization of XY is prechosen. Previous work on this problem includes both strictly elementary treatments and deeper, non-elementary approaches.The most common type of non-elementary approach uses results on S-unit equations to obtain a bound which is exponential in s, where s is the number of primes dividing XY c. A familiar general result of Evertse [7] shows that there are at most exp(4s + 6) solutions to the equation x + y = z in coprime positive integers (x, y, z) each composed of primes from a given set of s primes. It follows Mathematics Subject Classification: 11D61.

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Section: Introductionmentioning

confidence: 61%

Section: Sharper Results For N ≤mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Two terms with known prime divisors adding to a power

Scott¹,

Styer²

2018

Publ. Math. Debrecen

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“…We consider the cases when p 2z + 16 is an odd power of a prime, in other words, p 2z + 16 = q k for some odd prime q and odd integer k. When k > 1, such Diophantine equation has only integer solution p z = 3, q = 5, and k = 2, c.f. [4], Lemma 5.5. But this case corresponds to the curve '15a3', having torsion subgroup Z/2Z ⊕ Z/4Z.…”

Section: Exceptional Case This Family Is Parametrised By the Followimentioning

confidence: 98%

“…, where E d is the quadratic twist of E by d. 4 Let E d be the quadratic twist of E by d. In particular, suppose E is defined by the Weierstrass equation…”

Section: 12mentioning

confidence: 99%

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

Byeon

Kim

Yhee

2019

Int. J. Number Theory

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Let E be an elliptic curve defined over Q of conductor N, let M be the Manin constant of E, and C be the product of local Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field in which each prime divisor of N splits, P K be the Heegner point in E(K), and X(E/K) be the Tate-Shafarevich group of E over K. Also, let 2u K be the number of roots of unity contained in K. In [11], Gross and Zagier conjectured that if P K has infinite order in E(K), then the integer u K ·C · M · (#X(E/K)) 1/2 is divisible by #E(Q) tors . In this paper, we show that this conjecture is true.

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On elliptic curves whose conductor is a product of two prime powers

Sadek

2013

Math. Comp.

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We find all elliptic curves defined over Q that have a rational point of order N, N ≥ 4, and whose conductor is of the form p a q b , where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro's conjecture holds for these elliptic curves.problem under the condition that E has a rational torsion point. For example,in [6] the elliptic curves with conductor p m , p is prime, and 2-torsion points were listed.It was shown in [5] that all elliptic curves with conductor 2 m p n where p ≡ 3 or 5 mod 8, p = 3, that have a rational point of order 2, are effectively determined under the truth of the conjecture of Ankeny-Artin-Chowla.It is worth mentioning that the complete list of elliptic curves with a prime conductor has already been produced. The following theorem gives this list explicitly.Theorem 1.1 (Theorem 5.3.2, [10]). Let E be an elliptic curve over Q with prime conductor p. Then either |∆ E | = p or p 2 , or else p = 11 and ∆ E = 11 5 , or p = 17 and ∆ E = 17 4 , or p = 19 and ∆ E = 19 3 , or p = 37 and ∆ E = 37 3 . In particular, ∆ E | p 5 .The elliptic curves in Theorem 1.1 turn out to satisfy Szpiro's conjecture which is stated below for the convenience of the reader.Conjecture 1.2. If E is an elliptic curve over Q, thenOne of the popular strategies to find elliptic curves E/Q with a given conductor is to solve certain Diophantine equations obtained by equating the discriminant of E to the product of powers of the prime divisors of the conductor.Mazur gave a complete classification of the torsion subgroup E tors (Q) of E(Q), see (Theorem 7.5, §8, Chapter VIII, [11]). More precisely, E tors (Q) is isomorphic to one of the following fifteen groups: Z/nZ, 1 ≤ n ≤ 12, n = 11; or Z/2Z × Z/2nZ, 1 ≤ n ≤ 4.Given that P ∈ E(Q)[m], m = 2, 3, it is known that there exist b, c ∈ Q such that the following Weierstrass equation defines an elliptic curve E b,c isomorphic to E E b,c : y 2 + (1 − c)xy − by = x 3 − bx 2 with the image of P being (0, 0). The discriminant ∆(b, c) of E b,c is given as follows:By taking m to be an integer in {4, 5, 6, 7, 8, 9, 10, 12}, one finds an explicit relation between b, c, see for example §2 of [7].In this article, we generalize Theorem 1.1 to elliptic curves whose conductors have two distinct prime divisors only. More precisely, we generate the list of all elliptic curves

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THE EXPONENTIAL DIOPHANTINE EQUATION AX2 + BY 2 = λkZ AND ITS APPLICATIONS

Cited by 6 publications

References 18 publications

Two terms with known prime divisors adding to a power

Two terms with known prime divisors adding to a power

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

On elliptic curves whose conductor is a product of two prime powers

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