I93 2 -] ON EULER'S TOTIENT FUNCTION

Journal of Number Theory, 2003

In this paper we prove two results. The first theorem uses a paper of Kim [8] to show that for fixed primes p 1 , .

Theoretical Mathematics & Applications, 2022

A Fermat number is a number of the form F n = 2 2 n + 1, where n is an integer ≥ 0. In this paper, we show [via elementary arithmetic congruences] the following two results (R.) and (R'.). (R.): For every integer n ≥ 3, F n − 1 ≡ 1 mod[j], where j ∈ {3, 5, 17}. (R'.): For every integer n > 0 such that n ≡ 2 mod[6], we have F n − 1 ≡ 16 mod[19]. Result (R.) immediately implies that for every integer d ≥ 0, there exists at most two primes of the form 2F n + 1 + 10d [in particular, for every integer d ≥ 0, the numbers of the form 2F n + 1 + 10d (where n ≥ 2) are all composites ]; result (R.) also implies that there are infinitely many composite numbers of the form 2 n +F n and for every r ∈ {−2, 16}, there exists only one prime of the form r + F n. Result (R'.) immediately implies that there are infinitely many composite numbers of the form 2 + F n. That being said, we use the result (R.) and a special case of a Theorem of Dirichlet on arithmetic progression to explain why it is natural to conjecture that for every r ∈ {0, 2}, there are infinitely many primes of the form r + F n .

2021

Let N be an odd perfect number. Let ω(N) be the number of distinct prime factors of N and let Ω(N) be the total number of prime factors of N . We prove that if (3, N) = 1, then 302 113ω(N)− 286 113 ≤ Ω(N). If 3 | N , then 66 25ω(N)−5 ≤ Ω(N). This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on ω(N) in terms of the smallest prime factor of N and establish new lower bounds on N in terms of its smallest prime factor.

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:

2017

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cram\\'{e}r's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $\\pi(x)$ as well.

arXiv: Number Theory, 2018

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}\omega - \frac{286}{113} \leq \Omega. $ If $3|N$, then $\frac{66}{25}\omega-5\leq\Omega.$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $\omega(N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor.

In this paper we collected problems, which was either proposed or follow directly from results in our papers.

Acta Arithmetica, 2007

It is proved that equation (1) with 4 ≤ k ≤ 109 does not hold. The paper contains analogous result for k ≤ 100 for more general equation (2) under certain restrictions.

Mathematics of Computation, 2007

Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α k j=1 q 2β j j , where p, q1, · · · , q k are distinct primes and p ≡ α ≡ 1 (mod 4). Define the total number of prime factors of N as Ω(N ) := α + 2 k j=1 βj . Sayers showed that Ω(N ) ≥ 29. This was later extended by Iannucci and Sorli to show that Ω(N ) ≥ 37. This was extended by the author to show that Ω(N ) ≥ 47. Using an idea of Carl Pomerance this paper extends these results. The current new bound is Ω(N ) ≥ 75.