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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Taxicab numbers.

I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other. The first is $1729$. I'm trying to figure out if there'...
6
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158 views

Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
6
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105 views

Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$

I'm looking for composite $n$ such that $$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$ Are there only finitely many? Can this be proved? This is Sloane's A070037 but there's not much information in the ...
6
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134 views

About the isomorphism $\operatorname{Br}(\mathbb{Q}_p)\cong \mathbb{Q}/\mathbb{Z}$

I've been reading the section about Brauer groups in Introduction to Modern Number Theory, and I couldn't quite understand how this isomorphism is defined. We start with a central simple algebra $A$, ...
6
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155 views

Number of primefactors in $ f(n,W) = \prod_{k=1}^W (p_k^n -1) \text{ where } p_k=Prime(k) $

I'm reviving an old fiddling, although I do not yet really see its benefit. Beginning with the eulerproduct for the zeta-function in the representation $\small \zeta(n)=\prod {1\over 1-p^{-n} } = \...
6
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429 views

Ramanujan Series

I have the following question involving the series of $1/\pi^3$: Can we find such expansions by using the one for $1/\pi^3$ with $1/\pi^4$ or $1/\pi^n$ etc? Note that $$\frac{1}{32}\sum_{n=0}^\infty ...
6
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348 views

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 ...
6
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428 views

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? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ $\...
6
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128 views

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, ...
5
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126 views

Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
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43 views

Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers)

I want to prove that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers). By the definition of inverse limit, we know that there is a ring homomorphism $\phi$ from $\mathbb Z_3$ to $\mathbb Z/...
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52 views

Sets of natural numbers which are almost closed under addition

I am interested in a classification of sets $A \subseteq \mathbb{N}$ such that for all $k \in A$, $d( A+k \cap \mathbb{N} \setminus A) = 0$ where $d$ is the asymptotic density and $A+k = \{n \in \...
5
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64 views

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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64 views

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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96 views

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 ...
5
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138 views

Sum of like powers equal to a power

It's not hard to prove that $$(1+2+3+\ldots+n)^2=1^3+2^3+\ldots+n^3$$ ( for example using induction ) A generalization of this is also known : $$(\sum_{d \mid n} \tau(d))^2=\sum_{d \mid n} \tau(d)^3$...
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53 views

About the sum $ s_n:=\sum_{k=1}^{n} \text{lcm}(k,n) $

I'm currently studying the following sum: $$ s_n:=\sum_{k=1}^{n} \text{lcm}(k,n) $$ I manipulated it as follows: $$\begin{align} s_n&=\sum_{k=1}^{n} \text{lcm}(k,n)\\ &=\sum_{d|n}{\sum_{(k,n)=...
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57 views

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 $j$...
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79 views

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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90 views

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 ...
5
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100 views

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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61 views

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 $n \in \Bbb N^+$ ...
5
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174 views

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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139 views

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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120 views

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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87 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
5
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159 views

Neukirch's motivation for $p$-adic numbers

I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows: "The idea originated from the observation ...
5
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113 views

How prove that for a given integer $k$ there exist infinitely many $n$ such that $S(2^n+n)+k<S(2^n)$?

Let $S(n)$ be the sum of digits of $n$. How prove that for a given integer $k$ there exist infinitely many $n$ such that $S(2^n+n)+k<S(2^n)$? I have modified this problem: Denote by $S(a)$ the ...
5
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183 views

Is there a quicker way to generate integers which are hard to factor than multiplying two large primes?

An easy way to generate an integer which is hard to factor is to find two large primes and multiply them. As a bonus, you know the factors. I'm interested in whether it's possible to find integers (...
5
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64 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
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187 views

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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60 views

Why are there palindromic subsequences at random among this sequence?

So I was thinking about the Goldbach conjecture and I rephrased it to myself as the following: Prove that every number lies halfway between two primes (or is itself prime.) Which is equivalent. ...
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180 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
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124 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate $\...
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106 views

Official name of Fermat's $x^2+3y^2$ theorem?

One of Fermat's more well-known claims is that for any prime number $p$, $p=x^2+3y^2\iff p\equiv 1\pmod 3$. Does this have an "official" name? (Another one which goes $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
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61 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
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76 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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63 views

Kloosterman sum and multiples of 16

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. How can ...
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81 views

Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
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223 views

Does this prime generating way generate all the prime numbers?

I've thought of the following algorithm to find the entire list of prime numbers: Take a prime number $p$ to your list. $1.$ Multiply all the numbers in your list and call the number you get $...
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55 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
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175 views

Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as $3^3(3^{a-...
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133 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
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110 views

Is it decidable whether the iterates of a polynomial map are bounded?

Let $f:\mathbb{Q}^n\to \mathbb{Q}^n$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates $f(...
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49 views

Do modular forms of higher weight occur in the proof of FLT?

Fermat's last theorem is a consequence of a statement about weight two modular forms. Going through the long proof, does one ever encounter modular forms of higher weight?
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279 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} 1&0&0&...
5
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76 views

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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91 views

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; ...
5
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142 views

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 (http://en.wikipedia.org/...
5
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104 views

Number of points on an affine hypersurface

I am curious to know the following and I would appreciate any help! Let $k$ be a finite field of $q$ elements. Let $X \subseteq \mathbb{A}^{n+1}_k $ be the affine hypersurface defined by $f = \sum_{i=...