PRIME DIVISORS ARE POISSON DISTRIBUTED

2007 ◽  
Vol 03 (01) ◽  
pp. 1-18 ◽  
Author(s):  
ANDREW GRANVILLE
Keyword(s):  
Prime Divisors ◽  
Prime Factors ◽  
Cycle Lengths ◽  
Almost All

We show that the set of prime factors of almost all integers are "Poisson distributed", and that this remains true (appropriately formulated) even when we restrict the number of prime factors of the integer. Our results have inspired analogous results about the distribution of cycle lengths of permutations.

On the Normal Growth of Prime Factors of Integers

1992 ◽  
Vol 44 (6) ◽  
pp. 1121-1154 ◽  
Author(s):  
J. M. De Koninck ◽  
I. Kátai ◽  
A. Mercier
Keyword(s):  
Prime Factor ◽  
Continuous Distribution ◽  
Normal Growth ◽  
Distribution Functions ◽  
Prime Divisors ◽  
Simple Argument ◽  
Prime Factors ◽  
Almost All ◽  
Number Of Prime Divisors ◽  
Class Of Functions

AbstractLet h: [0,1] → R be such that and define .In 1966, Erdős [8] proved that holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, He further obtained that, for every z > 0 and almost all n, and that where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function We also investigate the two functions , where, in each case, h belongs to a large class of functions.

scholarly journals Cycle Lengths in a Permutation are Typically Poisson

10.37236/1133 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Andrew Granville
Keyword(s):  
Prime Divisors ◽  
Normal Order ◽  
Cycle Lengths ◽  
Number Of Cycles ◽  
Turán Theorem ◽  
Almost All

The set of cycle lengths of almost all permutations in $S_n$ are "Poisson distributed": we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain "normal order" (in the spirit of the Erdős-Turán theorem). Our results were inspired by analogous questions about the size of the prime divisors of "typical" integers.

A Variation on the Algorithm for GCD and LCM

1982 ◽  
Vol 30 (3) ◽  
pp. 46
Author(s):  
Verna M. Adams
Keyword(s):  
Prime Numbers ◽  
Prime Divisors ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors

An algorithm sometimes presented for finding the least common multiple (LCM) of two numbers uses tbe technique of simultaneously finding the prime factors of the numbers. This technique is shown in figure 1. Both numbers are checked for divisibility by 2, then by 3, by 5, and so on. If the divisor does not divide one of the numbers, the number is written on the next line as shown in steps 4 and 5. This process continues until all numbers to the left and on the bottom are prime numbers, or it can be continued, as shown in figure 1, until the numbers across the bottom are all ones. The least common multiple is the product of all of the prime divisors. Thus, LCM (80, 72) = 24 · 32 · 5.

Some Remarks on Prime Factors of Integers

1959 ◽  
Vol 11 ◽  
pp. 161-167 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Asymptotic Density ◽  
Prime Factors ◽  
Long Time ◽  
Image Position ◽  
Almost All

Let 1 < a1 < a2 < … be a sequence of integers and let N(x) denote the number of a's not exceeding x. If N(x)/x tends to a limit as x tends to infinity we say that the a's have a density. Often one calls it the asymptotic density to distinguish it from the Schnirelmann or arithmetical density. The statement that almost all integers have a certain property will mean that the integers which do not have this property have density 0. Throughout this paper p, q, r will denote primes.I conjectured for a long time that, if e > 0 is any given number, then almost all integers n have two divisors d1 and d2 satisfying1

scholarly journals Orders of reductions of elliptic curves with many and few prime factors

2017 ◽  
Vol 13 (08) ◽  
pp. 2115-2134
Author(s):  
Lee Troupe
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Extreme Values ◽  
Complex Multiplication ◽  
Prime Divisors ◽  
Prime Factors

In this paper, we investigate extreme values of [Formula: see text], where [Formula: see text] is an elliptic curve with complex multiplication and [Formula: see text] is the number-of-distinct-prime-divisors function. For fixed [Formula: see text], we prove an asymptotic formula for the quantity [Formula: see text]. The same result holds for the quantity [Formula: see text] when [Formula: see text]. This asymptotic formula matches what one might expect, based on a result of Delange concerning extreme values of [Formula: see text]. The argument is worked out in detail for the curve [Formula: see text], and we discuss how the method can be adapted for other CM elliptic curves.

A note on weakly prime-additive numbers

2021 ◽  
pp. 1-4
Author(s):  
Jin-Hui Fang
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Prime Factors ◽  
Positive Integers

A positive integer [Formula: see text] is called weakly prime-additive if [Formula: see text] has at least two distinct prime divisors and there exist distinct prime divisors [Formula: see text] of [Formula: see text] and positive integers [Formula: see text] such that [Formula: see text]. It is easy to see that [Formula: see text]. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

scholarly journals Almost primes in almost all short intervals

2016 ◽  
Vol 161 (2) ◽  
pp. 247-281 ◽  
Author(s):  
JONI TERÄVÄINEN
Keyword(s):  
Short Intervals ◽  
Almost Primes ◽  
Prime Factors ◽  
Almost All ◽  
Positive Integers

AbstractLet Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E3 numbers, and almost all intervals [x,x + log3.51x] contain E2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E3 numbers is optimal up to the ϵ in the exponent. The theorem on E2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.

Linear Equations with Small Prime and Almost Prime Solutions

2008 ◽  
Vol 51 (3) ◽  
pp. 399-405
Author(s):  
Xianmeng Meng
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Equations ◽  
Prime Factors ◽  
Almost All ◽  
Sharp Lower Bound ◽  
Almost Prime

AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.

scholarly journals Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

2012 ◽  
Vol 2012 ◽  
pp. 1-4
Author(s):  
A. Pekin
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Prime Divisors ◽  
Imaginary Quadratic Fields ◽  
Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, whereg>1is an integer and the discriminant of such fields has only two prime divisors.

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