Related papers: On Excess of the Odious Primes
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave estimates for the number of these primes. In this article we look at this conjecture,…
We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…
We continue our recent work on averages for ternary additive problems with powers of prime numbers.
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
We discuss various recent advances on weak forms of the Twin Prime Conjecture.
In this paper, we make some conjectures on prime numbers that are sharper than those found in the current literature. First we describe our studies on Legendre's Conjecture which is still unsolved. Next, we show that Brocard's Conjecture…
We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…
We consider the immediate consequence of an arguable addition to the standard Deduction Theorems of first order theories.
In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…
We proved that there are infinitely many cousin primes.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
We give a counterexample to a recently conjectured variant of the Penrose inequality.
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
In this paper, we prove certain theorems about three consecutive primes.
For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…