The greatest prime factor of the integers in a short interval (III)

1993 ◽  
Vol 9 (3) ◽  
pp. 321-336 ◽  
Author(s):  
Jia Chaohua
Keyword(s):  
Prime Factor ◽  
Short Interval ◽  
Greatest Prime Factor

scholarly journals Square-full numbers in short intervals

1991 ◽  
Vol 110 (1) ◽  
pp. 1-3 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Positive Integer ◽  
Riemann Hypothesis ◽  
Prime Factor ◽  
Short Interval ◽  
Short Intervals ◽  
Image Position

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] thatBateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we havewhen and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

scholarly journals Sommes friables d'exponentielles et applications

2015 ◽  
Vol 67 (3) ◽  
pp. 597-638 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Exponential Sums ◽  
Prime Factor ◽  
Character Sums ◽  
Contour Integration ◽  
Point Method ◽  
Saddle Point Method ◽  
Greatest Prime Factor

AbstractAn integer is said to be y–friable if its greatest prime factor is less than y. In this paper, we obtain estimates for exponential sums over y–friable numbers up to x which are non–trivial when y ≥ . As a consequence, we obtain an asymptotic formula for the number of y-friable solutions to the equation a + b = c which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

scholarly journals On the greatest prime factor of $(ax^m + by^n)$

1980 ◽  
Vol 36 (1) ◽  
pp. 21-25 ◽  
Author(s):  
T. Shorey
Keyword(s):  
Prime Factor ◽  
Greatest Prime Factor

Perfect powers in values of certain polynomials at integer points

1986 ◽  
Vol 99 (2) ◽  
pp. 195-207 ◽  
Author(s):  
T. N. Shorey
Keyword(s):  
Prime Factor ◽  
Integer Points ◽  
Greatest Prime Factor ◽  
Image Position

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d1, …, dt with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equationPutso that ½ < vt ≤ ¾. If α > 1 and kα < m ≤ kl, then equation (1) implies thatfor 1 ≤ i ≤ t and hence

scholarly journals On the greatest prime factor of $n^2+1$

1982 ◽  
Vol 32 (4) ◽  
pp. 1-11 ◽  
Author(s):  
Jean-Marc Deshouillers ◽  
Henryk Iwaniec
Keyword(s):  
Prime Factor ◽  
Greatest Prime Factor
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