Some lesser known open problems/ conjectures in number theory

Question

What are some lesser known problems/ conjectures in number theory ( especially on prime numbers ) which have evolved in these recent years and didn't got much of attention, and obviously wasn't featured in any global mathematical mainstream ,

It includes equivalences of pre existing conjectures/ results...

Please let me know, thanks in advance

Answer 1

Benoit Cloitre has a paper on 10 conjectures in additive number theory, and Wikipedia also has a long list of open problems, in particular in number theory.

Answer 2

Conjecture. A Diophantine equation in two variables is decidable.

Björn Poonen has many interesting papers on this, see this MO question. His article Hilbert's Tenth Problem over rings of number-theoretic interest is a pleasure to read. Several related questions are mentioned, e.g., whether or not the set $$ S=\{n\in \Bbb Z\mid n=x^3+y^3+z^3 \text{ for some integers $x,y,z$}\} $$ is recursive or not.

The general case of a Diophantine equation has a negative answer, see Hilbert's tenth problem, given by Matiyasevich.

The above conjecture has much less attention than, say, the Riemann hypothesis.

Actually, concerning the representation by three cubes, there is another conjecture, see this MO-question.

Answer 3

Another conjecture, which is more related to prime numbers, is the following one:

Kummer–Vandiver conjecture: A prime $p$ does not divide the class number $h_K$ of the maximal real subfield $K=\mathbb {Q} (\zeta _{p})^{+}$ of the $p$-th cyclotomic field.

The conjecture was first made by Ernst Kummer in $28$ December $1849$ and $24$ April $1853$ in letters to Leopold Kronecker. As of $2019$, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare.

Answer 4

Since you asked for prime numbers, one example would be Firoozbakht's conjecture, which states that the function $$f(n)=\sqrt[n]{p_n}$$ is strictly decreasing $\forall n\in\Bbb N$.

I would say it is not too well known in terms of the number of publications; for example, the conjecture does not feature in MathWorld (Wolfram), and a simple Google search brings up less than nine pages of results. In addition, browsing in Google Scholar yields only $54$ results.