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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0
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1answer
33 views

Show that there exists values of $x$ whose first digit is not $1$

Let $x$ be a positive integer. Show that if $x, x^2, x^3, \dots, x^n$ all start with the same digit, and $n$ is a positive integer, there exist values of $x$ whose first digit is not $1$. I ...
0
votes
1answer
21 views

Cofinite difference sets of integers

Is there any subset $A$ of integers such that (1) $A-A\subsetneqq 2\mathbb{Z}$; (2) the complement of $A-A$ in $2\mathbb{Z}$ is finite? ($A-A=\{a_1-a_2: a_1,a_2\in A\}$, and $2\mathbb{Z}$ is the ...
-5
votes
2answers
43 views

ISI math B question [on hold]

Consider $n>1$ lotus leaves placed around a circle. A frog jumps from one leaf to another in the following manner. It starts from some selected leaf. From there it skips exactly one leaf in the ...
2
votes
2answers
72 views

Show $x^2 + y^2 + 1 = 0 \pmod m$, iff $\,m \pmod 4 \ne 0$.

Show that $x^2 + y^2 + 1 = 0$ $\pmod m$ has solutions iff $\,m \pmod 4 \ne 0$. I know hot to show that this equation has solutions if m = p It's easy to show "$=>$", but I'm completery ...
0
votes
1answer
29 views

$X$th digit from end of $(111\dots)^2$

How do we find the $73$rd digit from ending of $(111111\dots 1)^2$ where ones are repeated $2012$?
3
votes
0answers
24 views

Sum of integers and zêta functions

I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance: $$\sum_{m \leqslant X} m ...
9
votes
5answers
2k views

Prove the fractions aren't integers

Prove that if $p$ and $q$ are distinct primes then $\dfrac{pq-1}{(p-1)(q-1)}$ is never an integer. Is it similarly true that if $p,q,r$ are distinct primes then $\dfrac{pqr-1}{(p-1)(q-1)(r-1)}$ is ...
0
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0answers
47 views

There are $101$ positive integers that sum to $300$. Can we find a subset of these integers that sums to $100$? [duplicate]

We are given a set of $101$ positive integers that sum to $300$. Since summation of $101$ distinct numbers cannot be $300$, repetition among the $101$ positive integers exists. Can we choose a group ...
1
vote
0answers
33 views

Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
-1
votes
0answers
17 views

Enquiry on the Riemann $\xi$ function. [on hold]

The Riemann xi function, $\xi(s)$, is known to be real valued on the critical line $s=1/2 + it$ where $t$ is real. But is it also real valued when $t$ is complex, that is, of the form $a+bi$ for some ...
6
votes
0answers
44 views

A counterexample to $x^n + y^n = h^2 + nf^2$ implies $x + y = h'^2 + nf'^2$ in the integers

The Wikipedia page for Sophie Germain contains the following: In the same 1807 letter, Sophie claimed that if $x^n + y^n$ is of the form $h^2 + nf^2$, then $x + y$ is also of that form. Gauss ...
-1
votes
0answers
32 views

Prove that the set contains $\Phi$ elements

Let $x \in \{0,1,\ldots,n-1\}$ where $n$ is a positive squarefree number. If $x$ is relatively prime to $m$ and $\Phi$ is the number of such $x$, prove that the set $\{a | x^m \equiv a \pmod{n} ...
2
votes
1answer
48 views

Questions that SAGE, MAGMA can answer?

I practice theoretical mathematics and I know (almost) nothing about SAGE, MAGMA. I would like to know (in general) what type of questions can I ask SAGE to do? For example, I know that given an ...
3
votes
2answers
47 views

Does $p=x^2+4y^2$ imply that $x$ is a quadratic residue mod $p$?

Does $p=x^2+4y^2$ imply that $x$ is a quadratic residue mod $p$? I'm stuck on this problem. My attempt: We know that since $p$ is a sum of squares $p\equiv 1 (4)$. This means that ...
1
vote
0answers
55 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
4
votes
0answers
54 views

Show that any positive rational number can be expressed as $\frac{a^3+b^3}{c^3+d^3}$. [duplicate]

Show that any positive rational number can be expressed as $$\frac{a^3+b^3}{c^3+d^3}$$ Perhaps the statement means that for every two positive integers $m$ and $n$, there exists a $k$ such that ...
1
vote
2answers
42 views

axiom of continuity guarantees no gaps exist on the real axis?

I read this content at the bottom of this page just wonder why axiom of continuity could guarantee no gaps exist on the real axis? any proofs ?
0
votes
1answer
15 views

How to permute remainders of CRT between residue classes?

I want to know how can be permuted the remainders of the CRT. How to go from $a \equiv r1 \;(\bmod\; n_1)$ $a \equiv r2 \;(\bmod\; n_2)$ to $b \equiv r1 \;(\bmod\; n_2)$ $b \equiv r2 ...
1
vote
2answers
49 views

Finding the smallest composition of a natural number with limited basic set of summands

W.l.o.g. I have a set of natural numbers $$S = \{s_1, \ldots, s_n\}, \quad s_i \in \mathbb N$$ as well as an $x \in \mathbb N$ I would like to express as sum of $s_i$. How do I find the smallest ...
0
votes
1answer
21 views

Entry of $1-9$ in magic box

There are 9 slots to fill. Question ask us to fill it using $1-9$ each being used only once. But what I can see here is that $5th$ column must be filled with $1,2$ and $3$ but after $1,2$ and $3$ ...
0
votes
2answers
58 views

Using remainder theorem in Pythagoras theorem makes absurd results!

At first, I apologize for the title. I really couldn't find anything better than this. Now,we know, some integers $a,b,c$ (none of them are $0$) can be found so that $$a^2 = b^2 + c^2$$ Now,here,of ...
0
votes
1answer
28 views

Power Tower modulo sequence

A power tower, defined here is $a\uparrow\uparrow n$. ($a\uparrow\uparrow 2=a^a$, $a\uparrow\uparrow 3=a^{a^a}$, $a\uparrow\uparrow 2=a^{a^{a^a}}$, etc...) Is there a base $a$ such that ...
0
votes
0answers
55 views

A Product of Primes [closed]

Anyone ever seen a function like this: F(1)=1st prime, F(2)=2nd prime * 1st prime, F(n)=nth prime * F(n-1)?
5
votes
3answers
153 views

Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
1
vote
1answer
41 views

On a certain prime structure.

It is unknown whether there are infinite primes $p$ where $2p-1$ is also a prime. Is it known there are only finitely many primes $p$ such that both $q$ and $2p-1$ are primes where $p-1=2aq$ for any ...
0
votes
0answers
23 views

How can we find all solutions to a Pell-type equation?

Is it true that for solveable Pell-type equations, all solutions are given by: 1: Finding fundamental solution to Pell's equation 2: Find all solutions of the Pell-type equation less then the ...
2
votes
1answer
88 views

Find all solutions to the Diophantine equation $x^2-7y^2=-3$

I want to find all integer solutions of the equation $$x^2-7y^2=-3$$ I don't really know where to start... I tried the one trick I know which is to factor in some quadratic ring: ...
3
votes
0answers
23 views

Iterate Over Integer Partition Refinement in Sage

A partition of an integer $n$ is a non-decreasing list of positive integers summing to $n$. For example, $3$ can be partitioned as $1 + 1 + 1$, $1 + 2$ or just $3$, but $2 + 1$ is indistinct from $1 + ...
1
vote
1answer
31 views

Integral formula for the local L factor of a base changed automorphic representation

Let $\Bbb A$ the ring of rational adeles and let $\pi=\bigotimes_{p\leq\infty}\pi_p$ be an automorphic (cuspidal) representation of ${\rm GL}_2(\Bbb A)$. Fix a quadratic extension $K\supset\Bbb Q$. ...
1
vote
1answer
35 views

When does a binomial have repeated roots mod p?

Given a polynomial $f(x)=x^n+a$, and I have that $p$ does not divide $an$, can I show that $f(x)\pmod p$ has no repeated roots? I'm not sure how to proceed.
1
vote
0answers
26 views

The largest product of two n-digit numbers which is palindrome

Project Euler: 4 is stated as follows: Largest palindrome product A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 ...
2
votes
2answers
41 views

Difference of subsets of integers with $A-A=2 \mathbb{Z}\setminus \{-2k,2k\}$

Is there any subset $A$ of integers such that $A-A= 2\mathbb{Z}\setminus \{-2k,2k\}$, for some integer $k$? ($A-A=\{a_1-a_2: a_1,a_2\in A\}$, and $2\mathbb{Z}$ is the set of even integers.)
2
votes
1answer
45 views

zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : ...
0
votes
0answers
50 views

A question about square roots of quadratic residues.

Suppose $\mathbb{Z}_p^*$ ($p$ is a prime) is a cyclic group with generator $g$. We consider a subgroup $\mathbb{G}$ of $\mathbb{Z}_p^*$ with generator $h$ and order $q$, where $h = g^4~mod~p$ and ...
0
votes
0answers
18 views

Finding the number of integer points inside a sphere of radius R and dimension D centered at the Origin

I am writing a computer program to count the number of integer points inside a sphere of radius R and Dimension D centered at the origin. In essence, if we have a sphere of dimension 2 (circle) and ...
-1
votes
0answers
37 views

Power of 3 with all odd digits?

Is the largest power of 3 with all odd digits 9? I can't seem to find a power of 3 that contain all odd digits except 3 and 9. (I checked the first 200 powers, and there is at least one even digit.) A ...
1
vote
1answer
76 views

Given $x_m=4x_{m-1}-x_{m-2},\ x_1=1,\ x_2=3$

I friend told me that apart from trivial ones, the elements in this sequence never equal powers of 3: $$x_m=4x_{m-1}-x_{m-2},\ x_1=1,\ x_2=3.$$ Could you please help me to prove this?
0
votes
0answers
15 views

for a chen prime p, what is the size of factors of p+2

Suppose the twin prime conjecture fails. Then, by Chen's theorem, there are infinitely many primes $p$ s. t. $p+2$ is a product of exactly two primes. It would be nice to know that as $p$ grows, so ...
7
votes
1answer
83 views

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$

Consider the following iterative process. We start with a 2-element set $S_0=\{0,1\}$. At $n^{\text{th}}$ step $(n\ge1)$ we take all non-empty subsets of $S_{n-1}$, then for each subset compute the ...
0
votes
0answers
35 views

Prove that the numbers are different

Let $n$ be an integer that is not divisible by any square greater than $1$. Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n$. If a period $t$ of $x_m$ is ...
0
votes
0answers
10 views

To find out HCF of two numbers [duplicate]

Given that $(a,b)=1\ \text{and}\ p\ \text{is odd prime,then prove that }(a+b,\frac{a^p+b^p}{a+b})\ \text {is 1 or}\ p$. have no idea where to start.any hint please
1
vote
1answer
27 views

Proving that if gcd(m, n) = 1, and if d divides mn, then there exist unique numbers a and b such that a divides m, b divides n, and d = ab.

What do I know? If d | mn, there exist an integer k such that dk = mn. I also know that because gcd(m, n) = 1 there exist some integers x and y such that mx + ny = 1. I am having trouble to prove ...
3
votes
1answer
12 views

Separable polynomial with splitting field an unramified extension?

I am trying to prove a theorem and it seems that I need that an irreducible polynomial $f(x)$ that is separable over $\mathfrak{p}$ has its splitting field an unramified extension of ...
0
votes
0answers
20 views

Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in ...
3
votes
0answers
37 views

Pairwise distinct subsequence

Suppose there is an infinite sequence $S_n = (s_1, s_2, \dots )$ generated by a finite set of numbers $\{1, 2, \dots, n\}$. Given a number $m$ such that $m < n$, the subsequence $(s_i, s_{i+1}, ...
0
votes
1answer
27 views

About $\sum_{i\geq 1}\frac{1}{(n+i)_{n+1}}$ and $\sum_{i\geq 1}\frac{1}{i^2-i-1}$

I was playing around with Zeta function and changed it as following to find that $$\sum_{i=1}^{\infty} \frac{1}{i\cdot(i+1)\cdot(i+2)\cdot\ldots\cdot(i+n)} = \frac{1}{n\cdot n!}$$ ...
1
vote
3answers
70 views

Number theory contradiction

I was solving this question How many pairs of integers $(a,b)$ that satisfies the following conditions: $1\le a,b\le 42$ $a^9\equiv b^7\pmod{43}$ and I ran into a contradiction: We ...
0
votes
0answers
22 views

Explicit form of a generating function.

Let $q \geq p$ be natural numbers both larger than or equal to two. Let $u(z):=z^p+z^{p+1}+...+z^q$ and $p(z)=\frac{z u'(z)}{1-u(z)}$. Since $p(z)$ is rational, one can write (by the theory of ...
4
votes
1answer
61 views

Is a strong form of Goldbach conjecture equivalent of Generlized Riemann Hypothesis?

In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS He said that: "we show that if a strong form of Goldbach's conjecture is true then every ...
2
votes
1answer
107 views

Find all $x,y,z \in \mathbb{Z^{+}}$ such that $20^x+15^y=2015^z$

Find all $x,y,z \in \mathbb{Z^{+}}$ such that $20^x+15^y=2015^z$ I was checking modulo $4$ to see that $y,z$ must have the same parity. Then took two cases when both $y,z$ are even and when both ...