Related papers: Differentiating by prime numbers
We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…
We explain the concept of p-values presupposing only rudimentary probability theory. We also use the occasion to introduce the notion of p-function, so that p-values are values of a p-function. The explanation is restricted to the discrete…
This work is meant to demonstrate new class of prime numbers -- cyclic prime numbers, that can be derived from any prime number at certain numeric systems. Cyclic prime numbers are also related to the cyclic numbers and full reptend prime…
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…
The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…
In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive…
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…
We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
We present a generalization of a formula of higher order derivatives and give a short proof.
We Define moments of partitions of integers, and show that they appear in higher order derivatives of certain combinations of functions.
We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.
We propose a diagrammatic notation for matrix differentiation. Our new notation enables us to derive formulas for matrix differentiation more easily than the usual matrix (or index) notation. We demonstrate the effectiveness of our notation…
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
Let $\delta(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le…
Differentiation is a cornerstone of computing and data analysis in every discipline of science and engineering. Indeed, most fundamental physics laws are expressed as relationships between derivatives in space and time. However, derivatives…