For an integer k >= 2, let {F-n((k))}(n >= 2-k) be the k-generalized Fibonacci sequence which starts with 0, ... ,0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms....
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For an integer k >= 2, let {F-n((k))}(n >= 2-k) be the k-generalized Fibonacci sequence which starts with 0, ... ,0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c with at least two representations as a difference between a k-generalized Fibonacci number and a power of 3. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.
For an integer d >= 2 which is not a square, we show that there is at most one value of the positive integer X participating in the Pell equation X-2 - dY(2) = +/- 1 which is a product of two Fibonacci numbers, wit...
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For an integer d >= 2 which is not a square, we show that there is at most one value of the positive integer X participating in the Pell equation X-2 - dY(2) = +/- 1 which is a product of two Fibonacci numbers, with a few exceptions that we completely characterize. (C) 2019 Elsevier Inc. All rights reserved.
In a recent paper [7] the author considered the family of parametrized Thue equations F-a (X, Y) := Pi (n)(i=1) (X - P-i(a)Y) - Y-n = +/-1, a is an element of N for monic polynomials p(1),...,p(n) is an element of Z[a...
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In a recent paper [7] the author considered the family of parametrized Thue equations F-a (X, Y) := Pi (n)(i=1) (X - P-i(a)Y) - Y-n = +/-1, a is an element of N for monic polynomials p(1),...,p(n) is an element of Z[a] which satisfy degp(1) < ... < degp(n). Under some technical conditions it could be proved that there is a computable constant a(0) = a(0)(p(1),...,P-n) such that for all integers a greater than or equal to a(0) the only integer solutions (x, y) of the Diophantine equation satisfy /y/ less than or equal to 1. In this paper, we give an explicit expression for a(0) depending on the polynomials p(1),...,p(n).
Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets A, we prove an analogous result for the function pA(n) that counts partitions of ...
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Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets A, we prove an analogous result for the function pA(n) that counts partitions of n into terms belonging to A.
Let a, b, c be integers. In this paper, we prove the integer solutions of the equation ax(y) + by(z) + cz(x) = 0 satisfy max{vertical bar x vertical bar, vertical bar y vertical bar, vertical bar z vertical bar} <=...
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Let a, b, c be integers. In this paper, we prove the integer solutions of the equation ax(y) + by(z) + cz(x) = 0 satisfy max{vertical bar x vertical bar, vertical bar y vertical bar, vertical bar z vertical bar} <= 2 max{a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max{x, y, z} < exp(exp(exp(5))).
In this paper, we find all Padovan numbers which can be written as are difference of two repdigits. It is shown that all Padovan numbers which can be written as a difference of two repdigits are P-k is an element of {...
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In this paper, we find all Padovan numbers which can be written as are difference of two repdigits. It is shown that all Padovan numbers which can be written as a difference of two repdigits are P-k is an element of {2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 200, 3329}.
We describe a method for complete solution of the superelliptic Diophantine equation ay(p) = f(x). The method is based on Baker's theory of linearforms in the logarithms. The characteristic feature of our approac...
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We describe a method for complete solution of the superelliptic Diophantine equation ay(p) = f(x). The method is based on Baker's theory of linearforms in the logarithms. The characteristic feature of our approach las compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n - 1)/2 at most;in many cases this number can be reduced. Two examples with p = 3 and n = 4 are given.
Let b >= 2 be a fixed positive integer and let S(n) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S(n) in base b is at least c(0) logn/(log log n), where c(0) is...
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Let b >= 2 be a fixed positive integer and let S(n) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S(n) in base b is at least c(0) logn/(log log n), where c(0) is some positive constant depending on b and on the sequence of binomial sums. Our results include middle binomial coefficients (2n n) and Apery numbers A(n). The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Baker's theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers. (C) 2011 Elsevier Inc. All rights reserved.
We solve several multi-parameter families of binomial Thue equations of arbitrary degree;for example, we solve the equation 5(u)x(n) - 2(r)3(s)y(n) = +/- 1, in non-zero integers x, y and positive integers u, r, s and ...
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We solve several multi-parameter families of binomial Thue equations of arbitrary degree;for example, we solve the equation 5(u)x(n) - 2(r)3(s)y(n) = +/- 1, in non-zero integers x, y and positive integers u, r, s and n >= 3. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linearforms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.
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