# 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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### Certain permutations of the set of all Pythagorean triples

The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970: http://www.jstor.org/stable/3613860 I learned ...
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### Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$\zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
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### Generalizing Quadratic Reciprocity Law with Dilates

Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, ...
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### The Boolean Pythagorean triples problem, a $200$-terabyte proof, and $d=163$

I came across this interesting math article, "Computer cracks 200-terabyte maths proof" where one phrase caught my attention and I quote, "... all triples could be multi-coloured in integers up to ...
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### proof - Bézout Coefficients are always relatively prime

I had been researching over the Extended Euclidean Algorithm when I happened to observe that the Bézout Coefficients were always relatively prime. Let $a$ and $b$ be two integers and $d$ their GCD. ...
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### Increasing function $f(x)$ such that $f(\gcd(x,y))=\gcd(f(x),f(y))$

This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as multiplicative function that maps primes to a permutation thereof. However, if ...
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### Describe the integral solutions to this cubic equation.

Consider the following cubic equation in $c$: $c^3 - 3c^2(a+b) + 3c(a+b) -3ab(a+b)-3=0$ Does this equation have infinitely many integer solutions $(a,b,c)$ ? EDIT: My attempt was rerwriting it as a ...
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### Basis of Quaternion Algebras

I was reading some notes in which the definition of quaternion algebra is given as : For $a,b \in k^{*}$ ,we define $k$-algebra by generators and relations as follows: it has two genrators $i$ and ...
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### Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
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### Prime spiral with regular triangles

I wrote a little program that creates a Ulam like spiral, only with regular triangles instead of squares. The following image shows the way it is set up: In this set up all the uneven numbers will ...
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### On the fourth power $2^4 + 15045^4 + 26870^4 + 34090^4 = 37239^4$

The Diophantine equation, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4\tag1$$ can be solved with $x_1 = 0$ (Elkies), or as the title shows, with $x_1 = 2$ (Wroblewski). See link here. Q: Can $(1)$ be solved in ...
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### Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
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### How big are the factors of $2^kp - 1$?

Let $p$ be a prime number, $p > 3$. Does it always exists a $k \in \mathbb N, k \ge 1$ such that the prime factors of $2^kp - 1$ are all less then $p$? Thoughts Well we can easily see that ...
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### Integer part of $e^x$

Using wolfram alpha gives that $\lfloor e^{2015}\rfloor \equiv 1 \pmod 4$. Is it possible to prove this by manual ? Thank you.
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### Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
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### finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
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### English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
### How find this $aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$
Question let $m$ is positive numbers,and such $m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^{m-1}+2\times (m+1)^{m-2}+\cdots+(m-1)\times (m+1)+m$$(or see ...