On prime divisors of the binomial coefficient

… computational, and algebraic aspects: proceedings of …, 1998

A Diophantine m-tuple with the property D(n), where n is an integer, is defined as a set of m positive integers with the property that the product of its any two distinct elements increased by n is a perfect square. It is known that if n is of the form 4k + 2, then there does not exist a Diophantine quadruple with the property D(n). The author has formerly proved that if n is not of the form 4k + 2 and n ∈ {−15, −12, −7, −4, −3, −1, 3, 5, 7, 8, 12, 13, 15, 20, 21, 24, 28, 32, 48, 60, 84}, then there exist at least two distinct Diophantine quadruples with the property D(n).

Inventiones Mathematicae, 1984

International Journal of Number Theory, 2013

In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ+ be fixed. Does there exist a positive integer k such that [Formula: see text] The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that [Formula: see text] The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems.

Indagationes Mathematicae, 2006

2021

The purpose of this note is to prove several binomial-like formulas whose exponents are values of the function ω(n) counting distinct prime factors of n.

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. The Conjectures Bhargava factorials. For definitions and basic results dealing with Bhargava's factorial functions we refer to [3], [4], [5] and [8]. Briefly, let X ⊆ Z be a finite or infinite set of integers. Following [5], one can define the notion of a p-ordering on X and use it to define a set of generalized factorials of the set X inductively. By definition 0! X = 1. Whenever p a prime, we fix an element a 0 ∈ X and, for k ≥ 1, we select a k such that the highest power of p dividing k−1 i=0 (a k − a i) is minimized. The resulting sequence of a i is then called a p-ordering of X. As one can gather from the definition, p-orderings are not unique, as one can vary a 0. On the other hand, associated with such a p-ordering of X we define an associated p-sequence {ν k (X, p)} ∞ k=1 by ν k (X, p) = w p (k−1 i=0 (a k − a i)), where w p (a) is, by definition, the highest power of p dividing a (e.g., w 2 (80) = 16). One can show that although the p-ordering is not unique the associated p-sequence is independent of the p-ordering used. Since this quantity is an invariant it can be used to define generalized factorials of X by setting k! X = p ν k (X, p), (1) where the (necessarily finite) product extends over all primes p. Definition 1. [12]. An abstract (or generalized) factorial is a function ! a : N → Z + that satisfies the following conditions:

1996

IntechOpen's , 2023

In this work we have studied the prime numbers in the model P ¼ am þ 1, m, a>1∈ . and the number in the form q ¼ mam þ bm þ 1 in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cmð Þ¼ a mam þ 1 is a generalized Cullen number then ma m - ð Þ1 a ð Þ mod Cmð Þ a . In a second paper published in 2014, they also presented a test for Broth’s numbers in Form kpn þ 1 where k<p n . These results are basically a generalization of the work of W. Bosma and H.C Williams who studied the cases, especially when p ¼ 2, 3, as well as a generalization of the primitive MillerRabin test. In this study in particular, we presented a test for numbers in the form mam þ bm þ 1 in the form of a polynomial that highlights the properties of these numbers as well as a test for the Fermat and Mersinner numbers and p ¼ ab þ 1 a, b>1∈  and p ¼ qa þ 1 where q is prime odd are special cases of the number mam þ bm þ 1 when b takes a specific value. For example, we proved if p ¼ qa þ 1 where q is odd prime and a>1∈  where πj ¼ 1 q q j
 then Pq2 j¼1 πjð Þ Cmð Þ a qj1 q  a m ð Þ - χð Þ m,qam ð Þ mod p Components of proof Binomial the- orem Fermat’s Litter Theorem Elementary algebra.

Canadian Mathematical Bulletin, 1986

Let ω(n) denote the number of distinct prime divisors of a positive integer n. Then we define and where are primes and r ≥ 2. Similarly denote by the number of divisors of n and let be defined by where are the divisors of n. We prove that there exists constants A and B such that and

Mathematics and Statistics, 2022

In this work we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or P = (10x + 7)(10y + 3) where P is a given natural number that ends in one. This allows us to decide the primality of a natural number P that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension n, n ≥ 2 , in which the relation is obtained: a 1 + a 2 + • • • + a n = λ[a 2 1 + • • • + a 2 n ], where a i is a real number for i = 1, ..., n. When n = 3 or n = 2 we manage to address Fermat's Last Theorem and the equation x 4 + y 4 = z 4 , proving that both equations do not have positive integer solutions. For n = 2, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.