# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Generalisation of $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b)$ to $\gcd\left(\frac{a^n-b^n}{a-b},a^m-b^m\right)$?

We have the identity $$\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b).$$ (see here) This appears to be a quite useful result with various applications. I wonder whether there is ...
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### How is graph theory used to solve problems in number theory?

What are some applications of graph theory in number theory? How can a graph theory approach be useful to solving number theory problems? In general, is graph theory ever useful in making number ...
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### Can we use $n\log n$ instead of $n$-th prime?

Denote $\pi(x)$ be the number of primes $\leq x,$ $p(n)$ be the $n$-th prime number. We have $\pi(p(n))=n.$ It's well known that $$\pi(x)\sim \frac{x}{\log x} \\p(n)\sim n\log n.$$ Is it always ...
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### How to list the prime factorised natural numbers?

Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
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### Sets, that have $\operatorname{LCM}\left(|c_1|,\dots,|c_p|\right)=\sum_{k=1}^p |c_k|$

I found that the least common multiple of the sizes of conjugacy classes $c_k$ of the symmetric group $S_n$ is equivalent to $n!$ the order of the group. Equivalently the sum of all $c_k$ is also $n!$....
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### Coprime multiplicative orders modulo infinitely many primes

Is it true that there are infinitely many primes $p$ such that the multiplicative orders of $2$ and $3$ are coprime $\pmod{p}$? By this I mean their order in $(\mathbb{Z}/p\mathbb{Z})^*$. If the ...
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### About solutions to $x^2+y^2+z^2=8n+3,n\in \mathbb N$

As we know that $x^2+y^2+z^2=8n+3,(n\in \mathbb N)\tag{1}$ has integer solutions $x,y,z\in \mathbb N.$ If $k\in \mathbb N$ has at least one prime factor which is $\equiv 3 \mod 4,$ then we call $k$ ...
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### Goldbach for certain classes of $n$

The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) states that In 1975, Hugh Montgomery and Robert Charles ...
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### For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
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### What primes were “pending” at the time of Wiles's proof of FLT?

I would like to know what instances of Fermat's Last Theorem were pending at the time of Wiles's proof. More specifically: what families of irregular primes had been discarded as possible ...
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### One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2$ in the ...
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### A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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### Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
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### Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant $1-4p$...
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### An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
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Consider the following polynomials: \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ... 0answers 127 views ### Ramification index of infinite primes I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension L/K of number fields, and an ... 0answers 131 views ### How many solutions are there to 2^a + 3^b = 2^c + 3^d? Are there only finite quadruples of non-negative integers (a,b,c,d) that satisfy the following equation:2^a + 3^b = 2^c + 3^d \quad ?$$with a \neq c. I found these: 5 = 2^2 + 3^0 = 2^1 + ... 0answers 70 views ### How to prove W(k) is a complete discrete valuation ring? I am trying to prove the fact that the ring of Witt Vectors W\left(k\right) is a complete discrete valuation ring, where k is a perfect field of characteristic p , but I'm stuck. Theorem 2 ... 0answers 150 views ### even square numbers represented as two primes added together Can every even square number be written as the sum of 2 prime numbers? How do you prove the result? 0answers 129 views ### A 9 dimensional number system? Given x and y as sums of nine cubes. When we multiply them, xy may again be written as a sum of nine cubes, due to Wieferich and/or Kempner. Does this invent a 9 dimensional number system, just ... 0answers 159 views ### Twin Prime Powers What are all the possible triplets of numbers a, b, c such that a+2=b, a+4=c, and all 3 are prime powers (where one must be a power of 3)? I'm aware of the cases for when they are ... 0answers 202 views ### expressing \log(\left \lfloor x \right \rfloor!) in terms of zeta-zeros let \psi(x) be the second Chebyshev Function. By the definition of this summatory function, and the fundamental theorem of arithmetic, we have the identity:$$\log(\left \lfloor x \right \rfloor!)=\...
I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...