Greatest Prime Factor Research Articles

On prime factors of terms of binary recurrence sequences

We establish estimates from below for the greatest prime factor of the $n$th term of a non-degenerate binary recurrence sequence when the sequence belongs to a class of sequences which includes the Lucas sequences.

Some Observations on the Greatest Prime Factor of an Integer

Abstract We examine the multiplicity of the greatest prime factor in k-full numbers and k-free numbers. We generalize a well-known result on greatest prime factors and obtain formulas related with the Riemann zeta function.

Quadratic polynomials at prime arguments

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.

On the quotient between the greatest prime factor and the least prime factor

On the quotient between the greatest prime factor and the least prime factor

Logarithm of the greatest prime factor

In this article we prove asymptotic formulae for sums where appear logkP(n), where P(n) is the greatest prime factor of n. In this way we generalize a formula of N. G. De Bruijn. Our methods are very elementary, we only use the concept of integral of a continuous function. We do not use the theory of complex functions.

Lower bounds for the greatest prime factor of product of consecutive positive integers

Under Baker's explicit abc conjecture, we completely solve a conjecture of Hickerson when a product of two or more factorials is equal to n! for a given positive integer n. We also prove it unconditionally when n≤e80. The proofs depend on sharpening the lower bounds on the greatest prime factor of a product of consecutive positive integers.

Sommes friables d'exponentielles et applications

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.

On the greatest prime factor of ab + 1

We improve some results on the size of the greatest prime factor of the integers of the form ab + 1 where a and b belong to some general given finite sequences A and B with rather large density.

Propriétés multiplicatives des entiers friables translatés

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour resembles that of all integers less than $x$. Extending a result of Basquin, we estimate the mean value over shifted friable numbers of certain arithmetic functions when $(\log x)^c \leq y$ for some positive $c$, showing a change in behaviour according to whether $\log y / \log\log x$ tends to infinity or not. In the same range in $(x, y)$, we prove an Erd\"os-Kac-type theorem for shifted friable numbers, improving a result of Fouvry and Tenenbaum. The results presented here are obtained using recent work of Harper on the statistical distribution of friable numbers in arithmetic progressions.

Sums of greatest prime factors

Suppose that k ≥ 2 and m ≥ 1 are fixed positive integers. Let B(n, p) be the number of positive integers not exceeding n such that the prime p is their greatest prime factor. In this article we obtain asymptotic formulae for Ck,m(n )= � n k <p≤n p m B(n, p)

On divisors of Lucas and Lehmer numbers

Let un be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of un which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

Irreducibility of generalized Hermite-Laguerre polynomials

For a rational $q=u+\frac{\alpha}{d}$ with $u, \alpha, d\in \ACOBZ$ with $u\ge 0, 1\le \alpha<d$, $\gcd(\alpha, d)=1$, the \emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\alpha +(n-1+u)d)x^{n-1}+\cdots\\ &\quad+a_1\left(\prod^{n-1}_{i=1}(\alpha +(i+u)d)\right)x+a_0 \left(\prod^{n-1}_{i=0}(\alpha +(i+u)d)\right) \end{align*} where $a_0, a_1, \cdots, a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\in \{\frac{1}{3}, \frac{2}{3}\}$ and extend some of the earlier irreducibility results when $q$ of the form $u+\frac{1}{2}$. We also prove a new improved lower bound for greatest prime factor of product of consecutive terms of an arithmetic progression whose common difference is 2 and 3.

The Greatest Prime Factor and Recurrent Sequences

The Greatest Prime Factor and Recurrent Sequences

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression

We prove the irreducibility of ceratin polynomials whose coefficients are in arithmetic progression with common difference 2 . We use Sylvester type of results on the greatest prime factor of a product with terms in an arithmetic progression and irreducibility results based on Newton Polygons.

Series involving the least and the greatest prime factor of a natural number

We determine necessary and sufficient conditions satisfied by the real numbers a, b, c that the series ∑ n2 n

On the greatest prime factor of sides of a Heron triangle

A Heron triangle is a triangle whose sides a, b, c and area A = s(s − a)(s − b)(s − c) are integers. Here, s = (a + b + c)/2 is the semiperimeter of . The triangle is called reduced if gcd(a, b, c) = 1. For a positive integer n let P(n) be the largest prime factor of n with the convention that P(1) = 1. In [5], it was shown that P(A) → ∞ as max{a, b, c} → ∞ through triples which are sides of reduced Heron triangles. In [4], it was shown that also P(abc) → ∞ again as max{a, b, c} → ∞ through triples which are sides of reduced Heron triangles. Here, we improve this to: Theorem 1. P(bc) goes to infinity as max{a, b, c} goes to infinity through triples which are sides of reduced Heron triangles .

On the greatest prime factor of ab + 1

We prove that whenever $$ \mathcal{A} $$ and $$ \mathcal{B} $$ are dense enough subsets of {1, ..., N}, there exist a ∈ $$ \mathcal{A} $$ and b ∈ $$ \mathcal{B} $$ such that the greatest prime factor of ab + 1 is at least $$ N^{1 + |\mathcal{A}|/(9N)} $$ .

Almost perfect powers in consecutive integers (II)

Let k ≥ 4 be an integer. We find all integers of the form by l where l ≥ 2 and the greatest prime factor of b is at most k (i.e. nearly a perfect power) such that they are also products of k consecutive integers with two terms omitted.

Squares in products in arithmetic progression with at most one term omitted and common difference a prime power

It is shown that a product of k−1 terms out of k ≥ 7 terms in arithmetic progression with common difference a prime power >1 is not a square. In fact it is not of the form by 2 where the greatest prime factor of b is less than or equal to k. Also, we show that product of 11 or more terms in an arithmetic progression with common difference a prime power > 1 is not of the form by 2 where the greatest prime factor of b is less than or equal to p π(k)+2 .

The greatest prime divisor of a product of terms in an arithmetic progression

For an integer v > 1, we denote by co(v) and P(v) the number of distinct prime divisors of v and the greatest prime factor of v, respectively, and we put co(l) = O, P(1) = 1. Further we write zra(v) for the number of primes ~ k. Let d = 1. A well known theorem of Sylvester [15] states that