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

The Density of Redugible Integers

1949 ◽  
Vol 1 (3) ◽  
pp. 297-299 ◽  
Author(s):  
S. D. Chowla ◽  
John Todd
Keyword(s):  
Prime Factor ◽  
Greatest Prime Factor ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Positive Integers

Introduction. The concept of a reducible integer was introduced recently [3] : if P(m) denotes the greatest prime factor of m then n is said to be reducible if P ( 1 + n2) < 2n. The reason for the term is that reducibility is a condition necessary and sufficient for the existence of a relation of the form where the ƒi are integers and the ni positive integers less than n. J. C. P. Miller pointed out to us the regularity of the distribution of the reducible integers (less than 600).

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 a sequence of prime numbers

1968 ◽  
Vol 8 (3) ◽  
pp. 571-574 ◽  
Author(s):  
C. D. Cox ◽  
A. J. Van Der Poorten
Keyword(s):  
Prime Factor ◽  
Prime Numbers ◽  
Image Position

Euclid's scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p1 = 2 and pn+1 is the highest prime factor of p1p2…pn+1.

scholarly journals On sparsely totient numbers

1991 ◽  
Vol 33 (3) ◽  
pp. 350-358
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Multiplicative Structure ◽  
Euler Totient Function ◽  
Image Position

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient ifHere φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) thatand infinitely often

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
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