# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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### Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?

The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does. Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then: $\Gamma(s)-\Gamma(1-s)$ yields zeros at: ...
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### Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
I came on the following multiple integral while renormalizing elliptic multiple zeta values: $$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.... 5answers 4k views ### Cliques, Paley graphs and quadratic residues A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ... 4answers 8k views ### Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares? It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ... 4answers 3k views ### Are There Primes of Every Hamming Weight? That is, for every integer n \in \mathbb{Z}_{>0} does there exist a prime which is the sum of n distinct powers of 2? In this case, the Hamming weight of a number is the number of 1s in ... 4answers 4k views ### Good uses of Siegel zeros? The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ... 6answers 4k views ### Infinitely many primes of the form 2^n+c as n varies? At the time of writing, question 5191 is closed with the accusation of homework. But I don't have a clue about what is going on in that question (other than part 3) [Edit: Anton's comments at 5191 ... 0answers 893 views ### Groups generated by 3 involutions Let r(m) denote the residue class r+m\mathbb{Z}, where 0 \leq r < m. Given disjoint residue classes r_1(m_1) and r_2(m_2), let the class transposition \tau_{r_1(m_1),r_2(m_2)} be the ... 2answers 1k views ### A closed form for an integral expressed as a finite series of \zeta(2k+1), \pi^m and a rational? In this paper the following beautiful integral expression for \zeta(3) is derived:$$\zeta(3)=\frac{1}{7}\,\int_0^{\pi} x\,(\pi-x)\csc(x)\, dx In a comment at the end of this question, I ...
I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. ...