Results 1  10
of
26
On the number of solutions of simultaneous Pell equations
"... It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x 2 − az 2 =1, y 2 − bz 2 =1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solu ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x 2 − az 2 =1, y 2 − bz 2 =1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solutions, this result is not too far from the truth. If, further, u and v are nonzero integers with av − bu nonzero, then the more general equations x 2 − az 2 = u, y 2 − bz 2 = v are shown to have ≪ 2 min{ω(u),ω(v)} log (u  + v) solutions in integers, where ω(m) denotes the number of distinct prime factors of m and the implied constant is absolute. These results follow from a combination of techniques including simultaneous Padé approximation to binomial functions, the theory of linear forms in two logarithms and some gap principles, both new and familiar. Some connections to elliptic curves and related problems are briefly discussed.
On Integral Zeros of Krawtchouk Polynomials
 J. Comb. Theory Ser. A bf
, 1996
"... We derive new conditions for nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented. Keywords: Krawtchouk polynomials, integral roots, perfect codes, switching reconstruction, Radon transform. 3 Research ..."
Abstract

Cited by 21 (5 self)
 Add to MetaCart
We derive new conditions for nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented. Keywords: Krawtchouk polynomials, integral roots, perfect codes, switching reconstruction, Radon transform. 3 Research supported by the Guastallo Fellowship and a grant from the Israeli Ministry of Science and Technology. 1. Introduction The binary Krawtchouk polynomial P n k (x) (of degree k) is defined by the following generating function: 1 X k=0 P n k (x)z k = (1 0 z) x (1 + z) n0x : (1) Usually n is fixed, and when it does not lead to confusion it is omitted. The question of existence of integral zeros of Krawtchouk polynomials (or, that is essentially the same, existence of zero coefficients in the expansion of (1 0 z) x (1 + z) n0x ) arises in many combinatorial and coding theory problems. Let us state some of them. 1. Radon transform on Z n 2 [14]. Let f : Z n 2 ! R; then the R...
On the nonholonomic character of logarithms, powers, and the nth prime function
, 2005
"... We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularitybased methods and several naturally occurring sequences are proved to be nonholonomic.
Some Applications of Diophantine Approximation
 Math. Inst. Leiden, Report MI
, 2000
"... The paper gives a survey of some results on diophantine approximation (Sections 1 and 2) and their applications (Sections 3,4 and 5). Section 1 contains an introduction to the theory of linear forms in logarithms of algebraic numbers and Section 2 some results following from the Subspace Theorem ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
The paper gives a survey of some results on diophantine approximation (Sections 1 and 2) and their applications (Sections 3,4 and 5). Section 1 contains an introduction to the theory of linear forms in logarithms of algebraic numbers and Section 2 some results following from the Subspace Theorem. In Section 3 we consider the local behaviour of sequences of numbers composed of small primes and of sums of two such numbers and in Section 4 the transcendence of innite sums of values of a rational function and related sums.
On the exponent of the group of points on elliptic curves in extension fields
 Intern. Math. Research Notices
"... Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory
, 2009
"... Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accele ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the selfcontradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.
VANISHING AND NONVANISHING OF TRACES OF HECKE OPERATORS
"... Abstract. Using a reformulation of the EichlerSelberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. nonvanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that fo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Using a reformulation of the EichlerSelberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. nonvanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite. These results motivate the “generalized Lehmer conjecture”, that the trace does not vanish for even weights 2k ≥ 16 or 2k =12.
Application of logic to combinatorial sequences and their recurrence relations
 In Model Theoretic Methods in Finite Combinatorics, volume 558 189  Computer Science Department  Ph.D. Thesis PHD201211  2012 of Contemporary Mathematics
, 2011
"... 1. Sequences of integers and their combinatorial interpretations 2. Linear recurrences 3. Logical formalisms 4. Finiteness conditions ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
1. Sequences of integers and their combinatorial interpretations 2. Linear recurrences 3. Logical formalisms 4. Finiteness conditions