# Questions tagged [prime-numbers]

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.

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### Primality test for specific class of $N=8kp^n-1$

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### Asymptotic density of sums of consecutive primes

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### Conjectured primality test for specific class of $N=k \cdot 6^n+1$

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### Arithmetic progressions, given a prime

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### Pairs of integers whose product is one more or less than a prime

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### How we can characterize all positive integers, multiples of 4, that cannot be expressed as $(p_1-1)(p_2-1),\;\;p_1,\,p_2$ distinct primes

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### A variant of Turán–Kubilius inequality

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### Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?

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### A conjecture involving $P_n=\prod_{k=1}^np_k$

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### The existence of rational points [closed]

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### Improved upper bound for second moment of reduced residues modulo $q$?

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### Write $2n+1=p+q$ with $p$ prime and $q$ practical

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### Are there infinitely many primes of the form $x^2+(x+y)y^2$?

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### p2 - p1 = 2n for every 2n [closed]

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### Are twin primes the only solution to this equation?

**2**

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### Do prime pairs inbetween and equidistant from adjacent integer powers cover all the prime numbers?

**7**

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### Prime-like numbers that avoid Green-Tao? [duplicate]

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### Erdos multiplication table problem avoiding integers with too many distinct prime factors

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### Possible rearrangments of double products containing sine function : [closed]

**14**

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### A tree with prime vertices

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### Construction of weight function to satisfy condition on given functional

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### The equivalent proposition of Legendre's conjecture [closed]

**4**

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### “On the distribution of reduced residues” by Montgomery and Vaughan – missing careful argument wanted

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### A conjecture stronger than the Legendre conjecture about prime numbers [closed]

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### Divisors of shifted geometric progressions

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### How to prove there is infinite prime numbers of form $5n+3$ without Dirichlet theorem? [closed]

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### On existence of conjecture relating prime zeta function:

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### On the asymptotics of the Chebyshev psi function

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### What are the exceptional properties of Mersenne exponent for known largest prime? [closed]

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### A conjecture/algorithm about prime numbers [closed]

**4**

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### Is there a connection of prime numbers and Extreme Value Theory?

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### The number of admissible tuples with last element equal to $h_{k-1}$?

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### A conditional approach to twin prime conjecture

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### Iteration of a primeness-measuring function

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### Explicit Formula for $n$th prime in terms of Riemann zeros?

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### Is Brun's constant a period?

**4**

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### Error term in Mertens' third theorem

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### Are numbers which are the product of n primes more common than numbers which are the product of n-1 primes? [duplicate]

**4**

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### Reference / Survey for finite field analog number theory

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### How to explain this prime gap bias around last digits?

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### Another kind of primality related to tessellations by polygons

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### Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

**2**

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### How does one prove that the density of unusual numbers is $\ln 2$?

**3**

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### Primes in arithmetic progressions above a given threshold

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### Equation of the Chebyshev $\psi$ function

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### Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers

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### Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

**2**

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### quadratic residues and cubic polynomials [closed]

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### Some interesting experimental results about the distribution of primes

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