# Tagged Questions

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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### About consecutive integers covered by arithmetic progressions [on hold]

Help me please to solve the following problem. There are $n$ arithmetic progressions of the form: $$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$ Initial integer terms $x_i \geq 0$ are varying. ...
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### Number of fixed points in Zagier's involution (Fermat's Theorem) [on hold]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace$ having ...
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### GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$. If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
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### Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
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### What did Yu Jianchun discover about Carmichael numbers?

There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...
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### When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$. Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$? Question: 1.How ...
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### An upper bound for the number of prime numbers in non-linear progressions

Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$. Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$. Can we get an upper bound for $\pi_f(x)$?
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### The largest primes in the monster group construction

The last three prime numbers in the factorization of the order of the Fischer Griess friendly monster group are 47, 59, 71. (https://en.wikipedia.org/wiki/Monster_group) On the other hand, the monster ...
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NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime. I have following ...
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### Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....
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### $p$-simple integers from between $n$ and $n+p-1$

Let $\ p\$ be an arbitrary prime. Then an integer $\ s\$ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\$ is not divisible by any prime $\ q<p.\$ Could you prove my conjecture (or is it ...
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### Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational? The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.
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### Recursive construction of integers from Fermat Primes

Assume there are only finitely many Fermat primes. Fix a Fermat prime $F_t$ say $3$. What is in general known about numbers of form $F_{t,k}=2^kF_t +1$? Is the number of primes of form $F_{t,k}$ ...
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### Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
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### The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
Is there a set of triple-primes satisfying the following equation? $p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$ I have checked the first 11 numbers that no one satisfy the above ...