42 problems

Let $a_1, \ldots, a_t$ be the longest sequence for which … Probably $t = \pi(n)$. Can one even prove $t < (1 + o(1))\pi(n)$ or at least $t = o(n)$? This latest conjecture will prob…

A purely computational problem (this problem cannot be attacked by other means at present). Call a prime $p$ good if every even number $2r \leq p - 3$ can be written in the form…

Perhaps the following rather silly conjecture could be added. Is it true that the set of odd integers not of the form $2^k + p$ is the not necessarily disjoint union of an infinite…

I just discovered in it a forgotten conjecture of mine, which might still be of interest. Let $\{p_n\}$ be the sequence of consecutive primes. Is it true that…

One could ask the following (probably unattackable) problem. Is it true that there is an $r$ so that every integer is the sum of a prime and $r$ or fewer powers of 2.

Crocker [16] proved that there are infinitely many odd integers not of the form $2^{k_1} + 2^{k_2} + p$, but his proof only gives that the number of integers $\leq x$ not of the fo…

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [$p_{n+1} - p_n = d_n$.] Turán and I [15] proved that the inequalities $d_{n+1} > d_n$ and $d_{n+1} < d_n$ both…

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [...] $p_{n+1} - p_n = d_n$. [...] It seems likely that $d_n/\log n$ is everywhere dense in $(0, \infty)$. Ricci…

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [...] $p_{n+1} - p_n = d_n$. [...] Sharpening previous results of Backlund, Brauer-Zeitz, and Westzynthius, I pr…

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. S…

Let $q$ be a sufficiently large positive integer and let $a$ be an integer with $(a,q)=1$. Erdős–Odlyzko–Sárközy conjecture. There exist primes $p_1,p_2\leq q$ such that … This con…

For each positive integer $n$, consider the consecutive squares $n^2$ and $(n+1)^2$. Conjecture on primes between consecutive squares. There is always a prime strictly between…

Let $X$ be large, let $\delta$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $X$ be large, let $h$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $F_N(r)$ denote the empirical distribution of $q_{\min}\bmod \ell$, let $\pi_{\mathrm{perm}}(r)$ be the permitted-residue distribution, and let $C_{\mathrm{opt}}(k,\ell)$ be a…

Chowla's conjecture. The least prime $p\equiv a\hspace{-0.12cm}\oldpmod {q}$ satisfies

Elliott--Halberstam conjecture. The range of moduli in this estimate can be extended to $q\leqslant x^{1-\varepsilon}$.

Let $p_n$ denote the $n$th prime, and let $\varepsilon>0$. Poisson Tail Conjecture. For $1\leq H\leq(\log X)^{2-\varepsilon}$, the two gap-counting quantities satisfy … … For…

Let $\alpha$ be a fixed irrational number and let $\tau>0$. Consider primes $p$ satisfying … where $\norm{\cdot}$ denotes the distance from a real number to the nearest integer. Pr…

Let $W\geq 1$ be fixed, and let $b_1,b_2$ be fixed distinct integers coprime to $W$. For each prime $p$, let $\nu_p(W\cdot+b_1,W\cdot+b_2)$ be the number of distinct residue classe…

Prime-divisor counting conjecture. For all primes $\ell$ and $1<\alpha\leq2$,

Let $X$ be large, let $L\ll X$, and let $q\geq 1$. Define … Suppose that $A\gg 1$. Short-interval prime progression conjecture. For all but an $o_{A\to\infty}(1)$ proportion of…

Quadratic bound conjecture. One has

Let $P(a,q)$ denote the smallest prime congruent to $a$ modulo $q$, and let $L$ be an exponent for which there is an effectively computable absolute constant $C$ such that … The Ge…

Gaussian distribution conjecture.