Patrick Ingram - Academia.edu (original) (raw)

Papers by Patrick Ingram

Research paper thumbnail of Multiples of integral points on elliptic curves

If EEE is a minimal elliptic curve defined over ZZ\ZZZZ, we obtain a bound CCC, depending only on t... more If EEE is a minimal elliptic curve defined over ZZ\ZZZZ, we obtain a bound CCC, depending only on the global Tamagawa number of EEE, such that for any point PinE(QQ)P\in E(\QQ)PinE(QQ), nPnPnP is integral for at most one value of n>Cn>Cn>C. As a corollary, we show that if E/QQE/\QQE/QQ is a fixed elliptic curve, then for all twists E′E'E′ of EEE of sufficient height, and all torsion-free, rank-one subgroups GammasubseteqE′(QQ)\Gamma\subseteq E'(\QQ)GammasubseteqE′(QQ), Gamma\GammaGamma contains at most 6 integral points. Explicit computations for congruent number curves are included.

Research paper thumbnail of PRIMITIVE DIVISORS ON TWISTS OF FERMAT'S CUBIC

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 +v3 = m, wi... more We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 +v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor. ... Let C denote a twist of the Fermat cubic, ... This paper is devoted to proving the following theorem.

Research paper thumbnail of Diophantine analysis and torsion on elliptic curves

Proceedings of The London Mathematical Society, 2006

Research paper thumbnail of THE UNIFORM PRIMALITY CONJECTURE FOR

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is sh... more An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, uncon- ditionally. In the latter case, a uniform bound is obtained on the index of a prime term.

Research paper thumbnail of Algebraic divisibility sequences over function fields

We study the existence of primes and of primitive divisors in classical divisibility sequences de... more We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Research paper thumbnail of Multiples of integral points on elliptic curves

Journal of Number Theory, 2009

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the globa... more If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.

Research paper thumbnail of Primitive divisors in arithmetic dynamics

Mathematical Proceedings of The Cambridge Philosophical Society, 2008

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does n... more Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

Research paper thumbnail of Primitive divisors on twists of the Fermat cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m,... more We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

Research paper thumbnail of The uniform primality conjecture for elliptic curves

Acta Arithmetica, 2008

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is sh... more An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.

Research paper thumbnail of TORSION SUBGROUPS OF ELLIPTIC CURVES IN SHORT WEIERSTRASS FORM

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgro... more In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E/Q in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any ε > 0, all but finitely many curves

Research paper thumbnail of Multiples of integral points on elliptic curves

If EEE is a minimal elliptic curve defined over ZZ\ZZZZ, we obtain a bound CCC, depending only on t... more If EEE is a minimal elliptic curve defined over ZZ\ZZZZ, we obtain a bound CCC, depending only on the global Tamagawa number of EEE, such that for any point PinE(QQ)P\in E(\QQ)PinE(QQ), nPnPnP is integral for at most one value of n>Cn>Cn>C. As a corollary, we show that if E/QQE/\QQE/QQ is a fixed elliptic curve, then for all twists E′E'E′ of EEE of sufficient height, and all torsion-free, rank-one subgroups GammasubseteqE′(QQ)\Gamma\subseteq E'(\QQ)GammasubseteqE′(QQ), Gamma\GammaGamma contains at most 6 integral points. Explicit computations for congruent number curves are included.

Research paper thumbnail of PRIMITIVE DIVISORS ON TWISTS OF FERMAT'S CUBIC

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 +v3 = m, wi... more We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 +v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor. ... Let C denote a twist of the Fermat cubic, ... This paper is devoted to proving the following theorem.

Research paper thumbnail of Diophantine analysis and torsion on elliptic curves

Proceedings of The London Mathematical Society, 2006

Research paper thumbnail of THE UNIFORM PRIMALITY CONJECTURE FOR

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is sh... more An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang&amp;amp;amp;amp;amp;amp;#39;s conjecture, and over the rational function field, uncon- ditionally. In the latter case, a uniform bound is obtained on the index of a prime term.

Research paper thumbnail of Algebraic divisibility sequences over function fields

We study the existence of primes and of primitive divisors in classical divisibility sequences de... more We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Research paper thumbnail of Multiples of integral points on elliptic curves

Journal of Number Theory, 2009

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the globa... more If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.

Research paper thumbnail of Primitive divisors in arithmetic dynamics

Mathematical Proceedings of The Cambridge Philosophical Society, 2008

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does n... more Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

Research paper thumbnail of Primitive divisors on twists of the Fermat cubic

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m,... more We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

Research paper thumbnail of The uniform primality conjecture for elliptic curves

Acta Arithmetica, 2008

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is sh... more An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.

Research paper thumbnail of TORSION SUBGROUPS OF ELLIPTIC CURVES IN SHORT WEIERSTRASS FORM

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgro... more In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E/Q in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any ε > 0, all but finitely many curves