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For every prime $p_i>2$ choose a $k_i\ge p_i$ , $k_i \in \mathbb{N}$ and take the arithmetic progression $A_i=k_i+np_i$ $n \ge 0$ . Is there any choice of the $k_i's$ such that $|\mathbb{N} \backslash …

Consider the following question: 1) For a given natural number $a$, are there finitely or infinitely many natural numbers that are not of the form $anm \pm n\pm m$, where $m$ and $n$ range over po …

After the change of the year I realized, as everyone did, that $2018=2\times1009$ and of course $1009$ is a prime number. $2017$ is also a prime number. Furthermore $2019=3\times 673$ and $673$ is als …

We define recursively $p_1=2,p_2=3$ and $$p_{n}= \min_{(A,B)\in F_{n-1}}|A-B| $$ Where $$ \begin{split} F_n=\{(A,B) |&\gcd (A,B)=1,\quad |A-B| \not =1, \\\ &\text{both $A$ and $B$ are products of po …

I asked this question at Mathematics Stack Exchange but since I didn't got a satisfactory answer I decided to ask it here as well. We write a power of 3 in bits in binary representation as follows. F …

From Michailescu's theorem (Catalan's conjecture) we have that the only $a,b,m,n \in \mathcal{Z}^{+}$ with $m,n>1$ such that $a^{m} - b^{n} = 1$ are: $a=3$, $b=2$, $m=2$, $n=3$. 1) Is there an algor …

I am trying to find lower and upper bounds for the number of integers that are coprime in pairs in an interval of length n. What are the best bounds that we have? Is that true that in any interval o …

Let $M$ be a natural number $M>1$. For every prime $p_i$ not dividing $M$ take an arithmetic progression $A_i=k_i+np_i$ , $n\geq 0$ such that $k_i>p_i^2/M$. Is there any $M$ and some choice of the …

1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)? (One can say that we can have it as a collorary of …

maybe you could take the primes on the products in any powers you want , it is a thought that i had before some years and it is also a natural question that one can make reading Euclid's proof since s …

Take the first $n$ primes $p_1,...,p_n$ and the primorial $P_n$ .Denote by $p_i$ every prime bigger than $p_n$ and smaller than $P_n$. 1) Is that true that there always be a number in any interval of …

Take an infinite set of distinct primes and a (edit: or 2 , etc.) residue class for every prime. For exammple you can take all the primes bigger than some prime or the primes of a specific form (i.e. …

Let's say that a subset $A$ of $\mathbb{N}$ has the Goldbach property if every even number $\geq4$ is the sum of two numbers of $A$. Are there any results and examples of low density sets with these …

This could be helpful: "Certain other cases of Dirichlet's Theorem have been proved by elementary methods; in fact, elementary proofs have been found for general classes…. M. Brauer found a rather sim …

one should add this :There are infinitely many twin primes if and only if there are infinitely many natural numbers that are not of the form 6nm+/-n +/-m. Proof: Every number that is not a multiply of …