Some Results on Number Theory and Analysis

In this paper we present a method to get the prime counting function π(x) and other arithmetical functions than can be generated by a Dirichlet series, first we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by Dirichlet series, then we could find a solution for π(x) and () () n x a n A x ≤ = ∑ , solving [ ] 0 J δ φ = for a given functional J, so the problem of finding a formula for the density of primes on the interval [2,x], or the calculation of the coefficients for a given arithmetical function a(n), can be viewed as some " Optimization " problems that can be attacked by either iterative or Numerical methods (as an example we introduce Rayleigh-Ritz and Newton methods with a brief description) Also we have introduced some conjectures about the asymptotic behavior of the series () n n p x x p ≤ Ξ = ∑ =S n (x) for n>0 , and a new expression for the Prime counting function in terms of the Non-trivial zeros of Riemann Zeta and its connection to Riemman Hypothesis and operator theory.

Experimental Mathematics, 2000

2004

Some nonconstant polynomials with a finite string of prime values are known; in this paper, some polynomials of this kind are described, starting from Euler’s example (1772) P(x) = x²+x+41: other quadratic polynomials with prime values were studied, and their properties were related to properties of quadratic fields; in this paper, some quadratic polynomials with prime values are described and studied.

Mathematics and Statistics, 2022

In this article, given a number P ∈ N that ends in one and assuming that there are integer solutions (A; B) ∈ N × N for the equations P = (10x + 9)(10y + 9) or P = (10x + 7)(10y + 3) or P = (10x + 1)(10y + 1), the straight line was used passing through the center of gravity of the triangle bounded by the vertices (A; A), (B; A), (A; B). Considering A ≥ 25, we manage to divide the domain of the curve P = (10x + 9)(10y + 9) into two disjoint subsets, and using Theorem (2.2) of this article, we find the subset where the integer solution of the equation P = (10x + 9)(10y + 9) is found. Similar process is done when P = (10x + 1)(10y + 1), in case P is of the form P = (10x + 7)(10y + 3) or P = (10x + 3)(10y + 7). These curves are different and to obtain a process similar to the one carried out previously, we proceeded according to Observation 2.2. Our results allow minimizing the number of operations to perform when our problem requires to be implemented computationally. Furthermore, we obtain some conditions to find the solution of the equations:

In this paper, we will discuss two questions, the primality test for Fermat number Fn and several fundamental problems in transfinite number theory. For the primality test for Fermat number Fn, this paper introduces the method of primality test for Fermat number Fn and the Fermat factorization theorem. Meanwhile, the factorization theorem is used to write operation procedure, so that we can distinguish whether 2n+2t+1 is the divisor of Fn or not. For some of the fundamental problems in transfinite number theory, this paper introduces the concept of countable expansion, and then inspects countability of the set of real numbers from the aspect of expansion. Besides, we analyze the cardinal number of linear point set, deriving the fact that their cardinal number possesses invariance.

2003

1. The Smarandache, Pseudo-Smarandache, resp. Smarandache-simple functions are defined as ([7J, [6]) S{n) = min{rn EN: nlm!}, Z(n) = min {m.E N: nl m{n~ + 1)} , 5p (n) = min{m EN: p"lm!} for fixed primes p. The duals of Sand Z have been studied e.g. in (2], [5J, [6]: 5.(n) = max{m EN: m!ln}, { m(rn+1) } Z.(n) = max mEN: 2

The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the product of the reversals of $A$ and of $B$. A pair $(A,B)$ of positive integers is said to be polynomial if the product $A \times B$ can be performed without carry. In this paper, polynomial pairs are introduced to study palindromic pairs. While polynomial pairs are readily seen to be palindromic, the converse does not hold in general but is conjectured to be true when neither $A$ nor $B$ is a palindrome. Connection is made with classical topics in recreational mathematics such as reversal multiplication, palindromic squares, repunits, and Lychrel numbers.

Acta Mathematica Hungarica, 1994