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 ...

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### How to prove a Mersenne number is not pseudoprime to the base 3?

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**1**answer

### The power of a prime in the prime factorization of a factorial [on hold]

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### Decomposition of prime into $q \cdot r + s$, where $q,r,s$ are primes

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### About a pattern on prime numbers [on hold]

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### About inverse function of π(x) (the prime counting function) [closed]

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**1**answer

### Are there infinitely many primes of this form?

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**1**answer

### A specific Diophantine equation restricted to prime values of variables.

**3**

**1**answer

### Higher roots modulo prime complexity best algorithm

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**1**answer

### Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]

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**4**answers

### Is there an 11-term arithmetic progression of primes beginning with 11?

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**1**answer

### Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

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**2**answers

### Primality test for specific class of Proth numbers

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**1**answer

### Is it true that there are infinite palindromic primes that when squared give palindromic number? [closed]

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### $p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?

**7**

**1**answer

### Goldbach's conjecture for the Liouville function

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### Does this recurrence yields only prime numbers?

**4**

**1**answer

### Density of integers with a large rough divisor

**6**

**1**answer

### Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?

**10**

**1**answer

### How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?

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**1**answer

### Upperbounding a sum of Legendre-Symbols

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### Left-right discrepancy in a short interval containing k0 primes

**4**

**1**answer

### Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$

**2**

**2**answers

### Sum of a terms in a divergent series taken along indices the sum of whose reciprocal diverges. Can the sum converge?

**10**

**1**answer

### Spacing of fractions with prime denominator

**4**

**1**answer

### Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?

**3**

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### Cancellation in this exponential sum?

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**1**answer

### Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?

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### Sergei numbers : even integers n being a prime gap at least n times

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**1**answer

### Can this weakening of Polignac's conjecture be proven?

**12**

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### Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?

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**1**answer

### Prove: If $P_n$ is $n$-$th$ prime number then $P_{n+m} \ge P_n+P_m$

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### Primes approximated by eigenvalues?

**5**

**2**answers

### $\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?

**4**

**2**answers

### Example of concrete statement which requires probabilistic algorithm

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**2**answers

### Sign of permutation induced by modular exponentiation

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**2**answers

### Prime factors of a sequence of integers which differ by consecutive prime differences

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### Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$

**3**

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### A Jacobian pair $(p,q)$ such that $\gcd(\deg(p),\deg(q))=2P$, $P \geq 5$ is prime

**2**

**1**answer

### Sum of reciprocals of integers minus primes

**1**

**1**answer

### On the regularity of integer solutions of a simultaneous equation with consecutive prime coefficients

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**1**answer

### On the largest prime factor of $1+n^k$

**4**

**1**answer

### Are all the numbers $\pi(n^2)/n^2\ (n=1,2,3,\ldots)$ pairwise distinct?

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### Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?

**3**

**0**answers

### Number of prime differences

**2**

**1**answer

### Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture

**0**

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### Mobius function on values of an irreducible quadratic polynomial

**2**

**0**answers

### On primitive roots of the form $5^k+10^m$ with $k$ and $m$ nonnegative integers

**2**

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### What is the regularized sum of the following series (sum of all primes but spaced with zeros in place of non-primes)?

**2**

**0**answers

### Does each odd prime $p$ have a primitive root $g < p$ which is the sum of two central binomial coefficients?

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