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31
Arithmetic and Growth of Periodic Orbits
, 2001
"... Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, ..."
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Cited by 26 (9 self)
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Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, is the property that the sequence asympototically approximates the number of periodic points under some map. In both cases we discuss when a sequence can have that property. For exact realizability, this amounts to examining the range and domain among integer sequences of the paired transformations
Primes Generated by Elliptic Curves
, 2003
"... For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the xcoordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel’s Theorem t ..."
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Cited by 19 (9 self)
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For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the xcoordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel’s Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan’s famous taxicab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.
Elliptic curves and related sequences
, 2003
"... A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers ..."
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Cited by 19 (2 self)
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A Somos 4 sequence is a sequence (hn) of rational numbers defined by the quadratic recursion hm+2 hm−2 = λ1 hm+1 hm−1 + λ2 h2 m for all m ∈ Z for some rational constants λ1, λ2. Elliptic divisibility sequences or EDSs are an important special case where λ1 = h2 2, λ2 = −h1 h3, the hn are integers and hn divides hm whenever n divides m. Somos (4) is the particular Somos 4 sequence whose coefficients λi and initial values are all 1. In this thesis we study the properties of EDSs and Somos 4 sequences reduced modulo a prime power pr. In chapter 2 we collect some results from number theory, and in chapter 3 we give a brief introduction to elliptic curves. In chapter 4 we introduce elliptic divisibility sequences, describe their relationship with elliptic curves, and outline what is known about the properties of an EDS modulo a prime power pr (work by Morgan Ward and Rachel Shipsey). In chapter 5 we extend the EDS “symmetry formulae ” of Ward and Shipsey
PRIME POWERS IN ELLIPTIC DIVISIBILITY SEQUENCES
, 2005
"... Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes. ..."
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Cited by 10 (7 self)
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Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.
Common divisors of elliptic divisibility sequences over function fields
, 2004
"... Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility se ..."
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Cited by 9 (0 self)
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Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility sequence of R. Let P, Q ∈ E(k(T)) be independent points. We conjecture that and that deg ( gcd(DnP,DmQ) ) is bounded for m, n ≥ 1, gcd(DnP,DnQ) = gcd(DP,DQ) for infinitely many n ≥ 1. We prove these conjectures in the case that j(E) ∈ k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E) ∈ k, we show that deg ( gcd(DnP,DnQ)) is as large as n + O ( √ n) for infinitely many n ̸ ≡ 0 (mod p).
The sign of an elliptic divisibility sequence
, 2004
"... Abstract. An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular EDS, a typical case being Sign(Wn) = (−1) ⌊nβ ⌋ for an i ..."
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Cited by 6 (0 self)
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Abstract. An elliptic divisibility sequence (EDS) is a sequence of integers (Wn)n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of Wn for unbounded nonsingular EDS, a typical case being Sign(Wn) = (−1) ⌊nβ ⌋ for an irrational number β ∈ R. As an application, we show that the associated sequence of absolute values (Wn) cannot be realized as the fixed point counting sequence of any abstract dynamical system.
Sigma function solution of the initial value problem for Somos 5 sequences
, 2008
"... The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we ..."
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Cited by 6 (4 self)
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The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.