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Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

2,266 questions
Newest Active Bountied 1 Unanswered
14 votes
2 answers
790 views

Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$. Is there any infinite class of primes ...
Euro Vidal Sampaio's user avatar
  • 1,125
-5 votes
0 answers
35 views

Background: Define phase blocks B(k) = [t_k, t_{k+1}] where \theta(t_k) = k\pi (these coincide with Gram points). The Hardy Z-function satisfies Z(t_k) = (-1)^k \cdot \text{Re}(\zeta(\tfrac{1}{2}+it_k)...
Sachin Sharma Arts's user avatar
  • 1
1 vote
0 answers
131 views

By Størmer's theorem, for every fixed $k$, there are only finitely many $n$ such that the set of prime divisors of $n(n+1)$ is a subset of the first $k$ primes. The OEIS sequence A141399 tracks those $...
Euro Vidal Sampaio's user avatar
  • 1,125
2 votes
0 answers
108 views

Question: Among the 84 known maximal prime gaps, consecutive records only occur for $p=2$ and $p=3$. Is it possible that for all $p > 3$, record-breaking gaps are strictly isolated? Inquiry: I am ...
Đào Thanh Oai's user avatar
  • 2,452
3 votes
2 answers
999 views

Let $p(n)$ be the smallest prime divisor of $n$. Some time ago, I heard of the following conjecture: If $q$ is a prime with the property that $2^q-1$ is composite, then $p(2^q-1)$ also has this ...
Euro Vidal Sampaio's user avatar
  • 1,125
5 votes
1 answer
184 views

Let $f(n)$ be an integer function such that $$ f(n) = \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{i} \sum\limits_{k=1}^{j} [(3ijk - (ij + ik + jk)) = n]. $$ Here square bracket denotes Iverson bracket. I ...
user avatar
-4 votes
0 answers
96 views

Recall that Legendre's Conjecture states that for every positive integer $n$, there is always a prime $p$ such that $n^2 < p < (n + 1)^2$. I am currently working on a symmetrical approach to ...
Mohammad Aldebbeh's user avatar
  • 1
0 votes
0 answers
137 views
+50

Conjecture: Let $x$ and $y$ be integers such that $1 < x < y$ and $\gcd(x, y) = 1$. There always exists at least one prime $p$ of the form: $$p = n \cdot x + m \cdot y$$ where $n, m \ge 1$ are ...
Đào Thanh Oai's user avatar
  • 2,452
1 vote
0 answers
154 views

Let $m_p(n)$ be the number of times, counted with multiplicity, that the prime $p$ occurs in all Pratt trees of $n$. A Pratt tree for a prime $p$ is known. For a composite number $n$ I take this ...
Orges Leka's user avatar
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2 votes
1 answer
473 views

Define the cumulative prime sum as $S(N) = 1 + \sum_{p \leq N} p$. Let $\{p_m\}$ be the sequence of prime numbers, and define a monotonically increasing sequence of test points $t_m = \log S(p_m)$. ...
anthem's user avatar
  • 11
1 vote
0 answers
124 views

I am a 15-year-old math enthusiast from Yemen, investigating quadratic polynomials of the form $f(n) = n^2 + bn + c$ where $b$ is a negative odd integer and $c$ is a prime. My research focuses on the ...
Ahmed omer's user avatar
  • 19
-2 votes
0 answers
92 views

Let R(n) denote the number of representations of an even integer $n$ as $n = p + q$, with $p < q$ and $p, q$ primes. I computed $R(n)$ inside the Baker���Harman–Pintz window $n ∈ [N, N + N^{0.525}]$, ...
Willy's user avatar
  • 1
3 votes
0 answers
228 views

For a given positive integer $a>1$, a composite number $n$ is called a (Fermat) pseudoprime to base $a$ if $$a^{n-1}\equiv1\pmod{n}$$ I wonder whether the following two questions concerning ...
Tong Lingling's user avatar
  • 1,036
1 vote
1 answer
282 views

As usual, let $d_n=p_{n+1}-p_n$ be the $n$th prime gap. By the results of Yitang Zhang and the subsequent Polymath project, we know that, infinitely often, $d_n$ does not exceed 246, and therefore it ...
Euro Vidal Sampaio's user avatar
  • 1,125
1 vote
0 answers
105 views

Here is a question, which originally occurred to me while thinking about the Gilbreath conjecture, and it might be the case that this problem is already solved, or popularly known to be unsolved yet. ...
Aditya Guha Roy's user avatar
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