On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

scholarly journals On the smooth values of shifted almost-primes

2019 ◽  
Vol 15 (01) ◽  
pp. 1-9
Author(s):  
Xiaodong Lü ◽  
Zhiwei Wang ◽  
Bin Chen
Keyword(s):  
Lower Bound ◽  
Prime Factor ◽  
Almost Primes ◽  
Prime Factors ◽  
Consecutive Integers

Denote by [Formula: see text] (respectively, [Formula: see text]) the largest (respectively, the smallest) prime factor of the integer [Formula: see text]. In this paper, we prove a lower bound of almost-primes [Formula: see text] with [Formula: see text] such that [Formula: see text] for [Formula: see text]. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

scholarly journals Consecutive integers with no large prime factors

1976 ◽  
Vol 22 (1) ◽  
pp. 1-11 ◽  
Author(s):  
R. B. Eggleton ◽  
J. L. Selfridge
Keyword(s):  
Lower Bounds ◽  
Prime Factor ◽  
Typical Result ◽  
Prime Factors ◽  
Consecutive Integers

For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

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 On strings of consecutive integers with no large prime factors

1998 ◽  
Vol 64 (2) ◽  
pp. 266-276 ◽  
Author(s):  
Antal Balog ◽  
Trevor D. Wooley
Keyword(s):  
Integer Coefficients ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Binomial Polynomials

AbstractWe investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

Prime Factors of Consecutive Integers

1972 ◽  
Vol 79 (10) ◽  
pp. 1082-1089 ◽  
Author(s):  
E. F. Ecklund ◽  
R. B. Eggleton
Keyword(s):  
Prime Factors ◽  
Consecutive Integers

On the class number of the lpth cyclotomic number field

1991 ◽  
Vol 109 (2) ◽  
pp. 263-276
Author(s):  
Norikata Nakagoshi
Keyword(s):  
Number Field ◽  
Class Number ◽  
Prime Factor ◽  
Analytic Formula ◽  
Cyclotomic Number ◽  
Class Number Formula ◽  
Prime Factors ◽  
Number Formula ◽  
Genus Number ◽  
Analytic Class

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

scholarly journals XXVIII.—On a Proposition in the Theory of Numbers

1857 ◽  
Vol 21 (3) ◽  
pp. 407-409
Author(s):  
Balfour Stewart
Keyword(s):  
Prime Number ◽  
Prime Factor ◽  
Prime Factors ◽  
Theory Of Numbers

Problem. If p be one of the roots of the equation xm − 1 = 0, (not 1,) then (1 − p)(1 − p2) …. (1 − pm − 1) = m, provided m is a prime number.If m be not a prime number, and if , the same will hold for all roots p = p1α, where α is a number < m and prime to m. But for all roots p = p1α, where α, or one of its prime factors, is also a prime factor of m, the product (1 − p)(1 − p2) …. (1 − pm − 1) will be equal to 0.

ON PRIME FACTORS OF ODD PERFECT NUMBERS

2012 ◽  
Vol 08 (06) ◽  
pp. 1537-1540 ◽  
Author(s):  
PETER ACQUAAH ◽  
SERGEI KONYAGIN
Keyword(s):  
Prime Factor ◽  
Perfect Number ◽  
Prime Factors ◽  
Perfect Numbers

We prove that a prime factor q of an odd perfect number x satisfies the inequality q < (3x)1/3.

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

scholarly journals ON THE CONJECTURE OF JEŚMANOWICZ CONCERNING PYTHAGOREAN TRIPLES

2009 ◽  
Vol 80 (3) ◽  
pp. 413-422 ◽  
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Prime Factor ◽  
Pythagorean Triples ◽  
Prime Factors ◽  
Positive Integers

AbstractLet a,b,c be relatively prime positive integers such that a2+b2=c2 with b even. In 1956 Jeśmanowicz conjectured that the equation ax+by=cz has no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b. The main results of this paper include the case where 8 divides b. One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors of b. Furthermore, we examine parities of the three variables x,y,z, and give new triples a,b,c such that the conjecture holds for the case where b is divisible by 8. In particular, to prove our results, we shall show an important result which asserts that if x,y,z are all even, then x/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.

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