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What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the…
It is well known that $\sum_{p\le n} 1/p =\ln(\ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge \ln(\ln(n)) + O(1)$. This is based on Euler's proof that $\sum_p 1/p$…
In this paper we investigate some interesting formulae of q-Euler numbers and polynomials related to the modified q-Bernstein polynomials.
In this paper we disprove a conjecture about numerators of divided Bernoulli numbers $B_n/n$ and $B_n/n(n-1)$ which was suggested by Roland Bacher. We give some counterexamples. Finally, we extend the results to the general case.
In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.
We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.
We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…
Carlitz has introduced an interesting $q$-analogue of Frobenius-Euler numbers in [4]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the $q$-Euler numbers. In this paper we give…
The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…
Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and…
When people mention the mathematical achievements of Euclid, his geometrical achievements always spring to mind. But, his Number-Theoretical achievements (See Books 7, 8 and 9 in his magnum opus \emph{Elements} [1]) are rarely spoken. The…
In this paper, we will define general Eulerian numbers and Eulerian polynomials based on general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian…
Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with…
Nous rappelons l'historique de la demonstration de la conjecture des fibres de Seifert, ainsi que ses motivations et ses diverses generalisations. ----- We recall the history of the proof of the Seibert fiber space conjecture, as well as…
We study the explicit formula of Euler numbers and polynomials of higher order
We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and…
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…
We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version…
I'll discuss how Goedel's paradox "This statement is false/unprovable" yields his famous result on the limits of axiomatic reasoning. I'll contrast that with my work, which is based on the paradox of "The first uninteresting positive whole…