Are there infinitely many Lucas primes
v1.3 research notesAre there infinitely many Lucas primes?...
Are there infinitely many Newman–Shanks–Williams primes
v1.3 research notesAre there infinitely many Newman–Shanks–Williams primes?...
Are there infinitely many Pell primes
v1.3 research notesAre there infinitely many Pell primes?...
Are there infinitely many Pierpont primes
v1.3 research notesAre there infinitely many Pierpont primes?...
Are there infinitely many prime quadruplets
v1.3 research notesAre there infinitely many prime quadruplets?...
Are there infinitely many prime triplets
v1.3 research notesAre there infinitely many prime triplets?...
Siegel's conjecture
v1.3 research notesSiegel's conjecture: are there infinitely many regular primes, and if so is their natural density as a subset of all primes $e^{-1/2}$?...
Are there infinitely many sexy primes
v1.3 research notesAre there infinitely many sexy primes?...
Are there infinitely many Wagstaff primes
v1.3 research notesAre there infinitely many Wagstaff primes?...
Are there infinitely many Wieferich primes
v1.3 research notesAre there infinitely many Wieferich primes?...
Are there infinitely many Wilson primes
v1.3 research notesAre there infinitely many Wilson primes?...
Are there infinitely many Wolstenholme primes
v1.3 research notesAre there infinitely many Wolstenholme primes?...
Are there infinitely many Woodall primes
v1.3 research notesAre there infinitely many Woodall primes?...
Can a prime p satisfy $2^{p-1}\equiv 1\pmod{p^2}$ and $3^{p-1}\equiv 1\pmod{p^2}$ simultaneously
v1.3 research notesCan a prime p satisfy $2^{p-1}\equiv 1\pmod{p^2}$ and $3^{p-1}\equiv 1\pmod{p^2}$ simultaneously?...
Does every prime number appear in the Euclid–Mullin sequence
v1.3 research notesDoes every prime number appear in the Euclid–Mullin sequence?...
What is the smallest Skewes's number
v1.3 research notesWhat is the smallest Skewes's number?...
Wikipedia number-theory item 171: For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pa…
v1.3 research notesFor any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fi...
For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})
v1.3 research notesFor any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})?...
Wikipedia number-theory item 173: For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are…
v1.3 research notesFor any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are there infinitely many repunit primes to base b?...
Wikipedia number-theory item 174: For any given integers $k\geq 1, b\geq 2, c\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there…
v1.3 research notesFor any given integers $k\geq 1, b\geq 2, c\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form $(k\times b^n+c...
Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$
v1.3 research notesIs every Fermat number $2^{2^n} + 1$ composite for $n > 4$?...
Is 509,203 the lowest Riesel number
v1.3 research notesIs 509,203 the lowest Riesel number?...
Pollock's octahedral-number conjecture
v1.3 research notesIs every positive integer expressible as a sum of at most seven octahedral numbers?...
Greenberg's pseudo-null conjecture
v1.3 research notesLet $F$ be totally real, let $\widetilde F$ be the compositum of all $\mathbb Z_p$-extensions of $F$, let $\widetilde L$ be its maximal unramified abe...
Second Hardy–Littlewood zeta-function conjecture
v1.3 research notesFor every $\varepsilon>0$, do constants $T_0(\varepsilon),c(\varepsilon)>0$ exist such that, for $T\geq T_0$ and $H=T^{1/2+\varepsilon}$, the number $...
7.1 (Long) — Principal and Euclidean rings of integers from totally real polynomials
v1.3 research notesLet $f(x)\in\mathbb{Z}[x]$ be irreducible over $\mathbb{Q}$ with all roots real, let $f(\alpha)=0$, let $k=\mathbb{Q}(\alpha)$, and let $\mathcal{O}_k...
7.2 (Long) — Clique numbers in unit- and prime-difference graphs
v1.3 research notesFor $\mathcal{O}_k$ as in item 7.1, let $\Gamma_{\mathrm{unit}}$ have vertex set $\mathcal{O}_k$, joining two elements when their difference is a unit...