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Fast Bounds on the Distribution of Smooth Numbers
, 2006
"... Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our fi ..."
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Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.
Multivariate Diophantine equations with many solutions
, 2001
"... Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + ..."
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Among other things we show that for each ntuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n Sunits are not contained in fewer than exp((4 + o(1))s 1=2 (log s) 1=2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for Sunit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 m < n, and any sufficiently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this paper, is an effective lower bound for the number K;T (X; Y ) of ideals in a number field K of norm X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm Y . This generalizes results of Caneld, Erdős and Pomerance [6] and of Moree and Stewart [19].
RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU
, 1887
"... to record his discoveries in notebooks in about 1904 when he entered the Government College of Kumbakonam for what was to be only one year of study. For the next five years, Ramanujan did mathematics, mostly in isolation, while logging his findings without proofs in ..."
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to record his discoveries in notebooks in about 1904 when he entered the Government College of Kumbakonam for what was to be only one year of study. For the next five years, Ramanujan did mathematics, mostly in isolation, while logging his findings without proofs in
A PAIR OF DIFFERENCE DIFFERENTIAL EQUATIONS OF EULERCAUCHY TYPE
"... Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory an ..."
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Abstract. We study two classes of linear difference differential equations analogous to EulerCauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general EulerCauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity. 1.