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12
Solving exponential Diophantine equations using lattice basis reduction algorithms
 J. NUMBER THEORY
, 1987
"... Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0 < x y < y” in x, y E S for Iixed 6 E (0, 1), and for the diophantine equation x + y = z in x, y, 2 G S. The method is based on multidimensi ..."
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Cited by 16 (2 self)
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Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0 < x y < y” in x, y E S for Iixed 6 E (0, 1), and for the diophantine equation x + y = z in x, y, 2 G S. The method is based on multidimensional diophantine approximation, in the real and padic case, respectively. The main computational tool is the L³Basis Reduction Algorithm. Elaborate examples are presented.
Diophantine approximations, Diophantine equations, Transcendence and Applications
 Indian J. Pure Appl. Math
"... This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c> 0 depending ..."
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Cited by 4 (1 self)
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This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c> 0 depending only on α such that  α − p c q qn for all rational numbers p q with q> 0. This inequality is from diophantine approximations. Any nontrivial improvement of this inequality shows that certain class of diophantine equations, known as Thue equations, has only finitely many integral solutions. Also, the above inequality can be applied to establish the transcendence of ∞ ∑ 1 numbers like. For an other example on connection between these topics, we 2n! n=1 refer to an account on equation (2) in this article.
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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Cited by 3 (1 self)
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
PRIME FACTORS OF ARITHMETIC PROGRESSIONS AND BINOMIAL COEFFICIENTS
"... Sylvester [Syl] proved in 1892 that a product of k consecutive positive integers x, x + 1,..., x + k − 1 greater than k is divisible by a prime exceeding k. It is a generalization of Bertrand’s Postulate that there is a prime among k+1, k+2,..., 2k. (Take x = k + 1.) The assumption x> k can not be r ..."
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Cited by 1 (0 self)
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Sylvester [Syl] proved in 1892 that a product of k consecutive positive integers x, x + 1,..., x + k − 1 greater than k is divisible by a prime exceeding k. It is a generalization of Bertrand’s Postulate that there is a prime among k+1, k+2,..., 2k. (Take x = k + 1.) The assumption x> k can not be removed since x = 1 should be
On Wieferich primes and padic logarithms ∗†
, 2008
"... We shall make a slight improvement to a result of padic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x p−1 − 1. 1 ..."
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Cited by 1 (1 self)
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We shall make a slight improvement to a result of padic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x p−1 − 1. 1
ON THE GREATEST AND LEAST PRIME FACTORS OF n!+l
"... Much work has been done on obtaining estimates from below for the greatest prime factors of the terms of certain sequences of integers. Let P(W) denote the greatest prime factor of nz and letf(x) be any irreducible polynomial with degree> 1 and integer coefficients. It can easily be deduced from Sie ..."
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Much work has been done on obtaining estimates from below for the greatest prime factors of the terms of certain sequences of integers. Let P(W) denote the greatest prime factor of nz and letf(x) be any irreducible polynomial with degree> 1 and integer coefficients. It can easily be deduced from Siegel’s work [ & 121 that P(f (x)) + w as x 3 co and recently Sprindzhuk and Kotov [7], using deep techniques of Baker, have shown that indeed P(f (x))> c log log x for all integers x where L ’ = c(f)> 0; the case of quadratic and cubic f was in fact covered by earlier works of Schinzel [lo] and Keates [6]. In another context Birkhoff and Vandiver Cl] proved, by elementary methods, that for distinct positive integers a, b, P(a”b”)> nf 1 for all integers 12> 6. Recently, again using techniques of Baker, the second author [14] obtained some new results in this connexion; for instance, for the Fermat numbers we have (see [15]) P(Y”$
unknown title
, 2008
"... ”Minoration effective de la distance padique entre puissances de nombres algébriques ” ∗†‡ ..."
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”Minoration effective de la distance padique entre puissances de nombres algébriques ” ∗†‡
ON PRIMITIVE DIVISORS OF n 2 + b
, 2007
"... Abstract. We study primitive divisors of terms of the sequence Pn = n 2 +b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult problem. We survey some results about divisors of th ..."
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Abstract. We study primitive divisors of terms of the sequence Pn = n 2 +b, for a fixed integer b which is not a negative square. It seems likely that the number of terms with a primitive divisor has a natural density. This seems to be a difficult problem. We survey some results about divisors of this sequence as well as provide upper and lower growth estimates for the number of terms which have a primitive divisor. 1. Primitive prime divisors Given b, an integer which is not a negative square, consider the integer sequence with nth term Pn = n 2 +b. It seems likely [3] that infinitely many of the terms are prime but a proof seems elusive. Perhaps this mirrors the status of the Mersenne Prime Conjecture, which predicts that the sequence with nth term Mn = 2 n − 1 contains infinitely many prime terms. At least with the Mersenne sequence, an old result shows that primes are produced in a less restrictive sense. Definition 1.1. Let (An) denote a sequence with integer terms. We say an integer d> 1 is a primitive divisor of An if (1) dAn and (2) gcd(d, Am) = 1 for all nonzero terms Am with m < n. In 1886 Bang [2] showed that if a is any fixed integer with a> 1 then the sequence with nth term a n − 1 has a primitive divisor for any index n> 6. This is remarkable because the number 6 is uniform across all a and it is small. Before we say any more about polynomials, a short survey follows indicating the incredible influence of Bang’s Theorem. 1.1. Primitive divisor theorems. In 1892 Zsigmondy obtained the generalization that for any choice of a and b with a> b> 0, the term a n − b n has a primitive divisor for any index n> 6. This lovely result was rediscovered several times in the early 20th century and it has turned out to be quite applicable. See [25] and the references
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"... On the greatest square free factor of terms of a linear recurrence sequence by ..."
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On the greatest square free factor of terms of a linear recurrence sequence by