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.

learn more… | top users | synonyms (1)

1
vote
2answers
27 views

Consecutive a-smooth numbers

I am looking for two large numbers $n, n+1$ such that both are $7$-smooth numbers. The two largest pairs I found are $2400, 2401$ and $4374, 4375$. Can anyone find a larger pair if it exists? Second, ...
0
votes
3answers
23 views

Decomposition of periodic functions

Suppose that $f$ is a periodic function defined on the integers with period $mn$, with $m$ and $n$ coprime integers. Does there necessarily exist a function $g$ with period $m$ such that $f-g$ is ...
3
votes
0answers
26 views

Subgroups of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are $Gal(\bar{\mathbb{Q}}/K)$, with ...
3
votes
1answer
18 views

Prove that it is sufficient to check $\lceil \log(k) \rceil$ pairs to tell if a set of integers is pairwise coprime

I am reading chapter 31 of Introduction to Algorithms (CRLS) and I encountered some difficulties while solving 31.2-9. I managed to prove the first part of a problem, but I can't prove the generalized ...
3
votes
2answers
69 views

Smallest gaps between powers of 2 and 3

I am trying to find the smallest gaps between powers of 2 and 3. Such examples are: (2, 3) (3, 4) (8, 9) (27, 32) (243, 256) (2048, 2187) (16384, 19683) (524288, 531441)... What are the next ...
-1
votes
0answers
17 views

Evaluation of a number at a big base. [on hold]

What are some special properties of an integer $N=a_nb^n+\dots+a_1b+a_0$ represented in base $b$ changing base to $b'\gg b$ while keeping coefficients same (that is properties of integer ...
1
vote
4answers
65 views

Databases for perfect numbers

So I have been trying to find a database that offers perfect numbers. I need this to help me and a friend with a project that we have been working on for a while involving the odd perfect number ...
1
vote
0answers
41 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 ...
1
vote
2answers
45 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
0answers
38 views

Prove the numbers are distinct

Let $n$ be a positive squarefree number. Prove that if $m$ and $p-1$ are relatively prime for every prime divisor $p$ of $n$, then the numbers \begin{align*}0^m &\equiv 0 \pmod{n} \\1^m ...
0
votes
1answer
778 views

Solving a Word Problem relating to factorisation [closed]

The $\text{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\text{Ionof}(18) = \frac{18}{6} = 3$, and ...
0
votes
0answers
14 views

Hardy- Littlewood Circle Method

I'm currently trying to get to grips with the Hardy Littlewood circle method so I'm working through Vaughan's book. In the past I've been very bad for leaving a point behind if I don't follow it so ...
0
votes
2answers
29 views

I don't understand the algorithm for solving equations of the form $x^n \equiv 1 \mod m$

Given a congruential equation of the form $x^n \equiv 1 \mod m$, according to my course notes all I have to do is to find a primitive root $a \mod m$ and then the solutions to the equation are of the ...
0
votes
0answers
14 views

Prove: $\bar{a}^2 = \bar{0}$ in $\mathbb{Z}_n \rightarrow \bar{a}=0$

For this summer, I am teaching myself abstract algebra and I've been working on a proof for the following statement. I just need someone to confirm whether it is sound. (Note: Here, $\bar{a}$ denotes ...
3
votes
3answers
51 views

Finding domain of $f \circ g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then $f ...
114
votes
9answers
12k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
-5
votes
2answers
40 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 ...
10
votes
0answers
304 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
2
votes
2answers
65 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
24 views

Looking for a simpler solution to a problem about the divisibility of combinatorial numbers

Here is the problem: For every positive integer r, there exists a natural number $n_r$ such that for every integer $n>n_r$, there is at least one $k$, where $1\leq k \leq n-1$,such that ...
0
votes
1answer
19 views

Dedekind psi function

How are you supposed to compute the Dedekind psi function?? Can someone give me an expansion that will explain how it works? I have already tried the wikipedia article on it, but I don't see what the ...
1
vote
1answer
16 views

$a$, $n$ are relatively prime iff least significant digit of representation $(a)_{n}$ base n is relatively prime to n

Please can someone help me with this: Prove that $a$ and $n$ are relatively prime iff $a_{0}$ is relatively prime to n, where $a_{0}$ is the last digit of the representation $a_{n}$ base $n$ Thank ...
1
vote
0answers
55 views

Is this a valid proof of Goldbach's conjecture? [on hold]

Goldbach's conjecture is true if and only if there exists a couple of prime numbers (p,p'), with p>p', such that the couple (1,-1) is a solution to the following equation: $px-p{}'y=2k, \forall k> ...
1
vote
2answers
27 views

In $GF(q)$, are there the same number of quadratic residues as quadratic nonresidues?

I know that for $\mathbb Z_p^*$ (the multiplicative group of a field with $p$ elements where $p$ is a prime), there are $(p-1)/2$ quadratic residues, and thus $(p-1)/2$ quadratic nonresidues. We can ...
3
votes
2answers
19 views

Constructing a set of integers coprime with $n$ with a certain property

Hello what I would like to do is take a fixed arbitrary integer $n$ (it is safe to assume it is fairly large if need be) and construct a set of integers, $P_i$, which are all coprime with $n$ such ...
8
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
votes
1answer
30 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 ...
1
vote
1answer
36 views

Any power of $x$ from $1$ to $n$, inclusive, starts with a $9$

For any positive integer $n$, prove that we may choose a sufficiently long string of $9$s for a positive integer $x$ so that any power of $x$ from $1$ to $n$, inclusive, starts with a $9$. I was ...
4
votes
1answer
1k views

Numbers that can be expressed as the sum of two cubes in exactly two different ways

It is true that there are many numbers which can be expressed as the sum of two cubes,in two different ways, but 1729 is the SMALLEST such number. (2 = 1^3+1^3 is not considered as both are 1^3). ...
0
votes
1answer
16 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 ...
1
vote
1answer
25 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
0answers
30 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 ...
4
votes
3answers
116 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 ...
0
votes
1answer
24 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 $2012$?
4
votes
1answer
58 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 ...
3
votes
0answers
22 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 ...
2
votes
1answer
43 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 ...
65
votes
10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
0
votes
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 ...
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
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
41 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} ...
3
votes
2answers
44 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
2answers
35 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 ?
4
votes
0answers
51 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 ...
2
votes
2answers
129 views

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham's number and before we can even start calculating G1 which is $3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of $3 ...
0
votes
1answer
20 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
57 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 ...
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.