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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21
votes
2answers
419 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
3
votes
4answers
137 views

Clustering numbers by factors count

Is there a formula (or an efficient approach) for counting amount of positive numbers in range up to $N$ which have exactly $K$ divisors? P.S. Initial problem was to cluster number in range [1..N] ...
2
votes
0answers
203 views

A good introduction to S unit equations

I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper. ...
11
votes
2answers
791 views

crazy problem - does it have a solution? number theory perhaps?

A few researchers are trying to crack a code which involves discovering the values of three integers. They know they are between 1 and 100 (inclusive), and that they may be the same. They each have a ...
8
votes
1answer
601 views

Finding the first number larger than N that is a relative prime to M

I am not sure if this question is best suited for math exchange. I already tried on stackoverflow without any luck, so I hope that your bright minds will be more helpful. So, basically the tile says ...
0
votes
1answer
106 views

A question on primitive roots

Let $p$ be an odd prime. How can Ihow that $a$ is a primitive root modulo $p$ iff $a^{(p-1)/q}\ncong 1 \pmod{p}$ for all prime divisors $q$ of $p-1$. Thanks
1
vote
1answer
203 views

Is a sum of equidistributed sequences $\sum \{n \alpha\}$ equidistributed?

We have two equidistributed sequences {n a} and {n b} (mod 1), in which a and b are irrational. 1) Is it true that the sum {na} + {nb} is equidistributed? and 2) Is it true that {na} + {nb} = ...
7
votes
9answers
1k views

prove for all $n\geq 0$ that $3 \mid n^3+6n^2+11n+6$

I'm having some trouble with this question and can't really get how to prove this.. I have to prove $n^3+6n^2+11n+6$ is divisible by $3$ for all $n \geq 0$. I have tried doing $\dfrac{m}{3}=n$ and ...
3
votes
2answers
157 views

Finding primes for which a given number is a perfect square.

Find all primes p such that $(2^{p-1}-1)/p$ is a perfect square. I know we can factorise $(2^{p-1}-1)$ into two distinct factors which will be coprime and hence p can divide one of these factors ...
0
votes
1answer
211 views

does $ab$ divide $(ab-1)!$?

i am interested in following thing, that for $a \gt 2$ , $b \gt2$, $ab$ divides $(ab-1)!$ ? I can take some simple example, for example $(3,3)$,or $(3,5)$ and show that this is true by ...
60
votes
1answer
3k views

Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and ...
3
votes
3answers
298 views

Simplest proof that $\zeta(s) \to \infty$ as $s \to 1$?

For homework I had to prove the divergence of the series $1/(k\log^p k)$ for all real $p$ (it is simple to do so via integration.) However a more elegant means would be to appeal to the behavior of ...
3
votes
1answer
657 views

split in cyclotomic field

$K=Q(\zeta_n)$ a cyclotomic extension: $p$ splits completely in $K$ if and only if $p\equiv 1\ (mod\ n)$ I don't know how i could prove, I search a kind of cyclotomic reciprocity law Many thanks
0
votes
1answer
89 views

key for opening doors

You are standing on the middle of a East-West street. A row of closed doors in front of you, and you have a key on hand which can only open one specific door. Compose a strategy so that X/N value is ...
3
votes
1answer
676 views

Generating Function for Even Number of Odd Parts

How I would write the generating function for a partition of a positive integer n with an even number of odd parts? Any hints or suggestions will be greatly appreciated.
12
votes
0answers
619 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
3
votes
1answer
181 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
2
votes
0answers
265 views

Product of all integers less than the largest 12 digit prime number

For a prime number $p$ the set of co-primes less than or equal to it is given by $\{1,2,3,4,...p-1\}$. For $0<x<p$, we define $f(x,p)= 1$ if and only if all the numbers from $1$ to $p-1$ can be ...
6
votes
1answer
111 views

Number of roots

Would I be right in thinking that $x^m\equiv 1 (\mod n)$ has only $m$ distinct roots? If not, would it be true if m,n are coprime or simply distinct primes? My gut feeling is that there should ony be ...
4
votes
1answer
276 views

Gauss-type sums for cube roots

(Quadratic) Gauss sums express square root of any integer as a sum of roots of unity (or of cosines of rational multiples of $2\pi$, if you will) with rational coefficients. But Kronecker-Weber ...
12
votes
4answers
502 views

Counting (Number theory / Factors)

I'm stuck with this counting problem: I have an expression: $T = (N!) \times (N!) / D$ where, $D \in [1 - N!]$, i.e. $D$ takes all values from $1$ to $N!$ and I'm to count the number of points where ...
3
votes
1answer
219 views

number of additive partition

I have a question related with number of additive partition or method similar like this: $$p(5)=1+4=2+3=1+1+1+1+1=1+1+1+2=1+2+2=1+1+3$$ For a given number $n$,if we are trying to calculate ...
1
vote
1answer
361 views

Verifying identities for Riemann zeta function

I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
2
votes
0answers
793 views

Computing the Minimum Number of Squares Needed to Sum to $n$

I am aware of Lagrange's Four-Square Theorem, which states that every positive integer can be written as the sum of at most four squares. Clearly some integers require fewer squares. Does there ...
2
votes
1answer
309 views

Number of primes less than 2n

A series of questions. Explanations would be useful. I have done the first four parts. I am confused on how to go about the last two. Show that for any prime $p$ the largest power of $p$ that ...
0
votes
1answer
113 views

the minimum polynomial of a unit

Let $A$ be a dvr of characteristic zero. Let $B/A$ be a finite integral extension of $A$. Suppose that there exists a unit $x$ in $B$ such that $B=A[x]$. What can we say about the minimal polynomial ...
1
vote
1answer
84 views

How do I compute the discrete valuation of the sum of two elements

Let $O$ be a discrete valuation ring with valuation $v$. We normalize by $v(\pi) =1$, with $\pi$ a prime element in $O$. By definition, for all $x,y$ in $O$, we have $v(x+y) \geq \min (v(x),v(y))$. ...
2
votes
2answers
127 views

Finding relatively prime integers with specified remainders $\pmod{m}$.

Let $m$ be a positive integer. Let $a,b$ be integers with $0 \leq a,b < m$, $a,b$ not both zero, $\gcd(a,b,m)=1$. Do there necessarily exist integers $x,y$ such that $x \equiv a \pmod{m}$ $y ...
4
votes
4answers
222 views

When is $a^k \pmod{m}$ a periodic sequence?

Let $a$ and $m$ be a positive integers with $a < m$. Suppose that $p$ and $q$ are prime divisors of $m$. Suppose that $a$ is divisible by $p$ but not $q$. Is there necessarily an integer ...
1
vote
1answer
89 views

Decreasing function made up of cubes, squares and floor function

Let $t$ and $r$ be two integers with $r\geq 1, t\geq \frac{r}{2}$. Put $$ f(r,t)=\lfloor 2(t^2+r)^{\frac{3}{2}}-(2t^3+3rt) \rfloor $$ (here $\lfloor x \rfloor$ denotes the floor of $x$, i.e. the ...
11
votes
3answers
1k views

Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
6
votes
1answer
484 views

Pythagorean triplets

Respected Mathematicians, For Pythagorean triplets $(a,b,c)$, if $c$ is odd then any one of $a$ and $b$ is odd. Here $(a, b, c)$ is a Pythagorean triplet with $c^2 = a^2 + b^2$. Now, I will ...
3
votes
1answer
2k views

Irreducible elements in $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?

First of all I am new to this topic, algebraic number theory, so I only know a decent (not great) amount of abstract algebra. The question I have is that, given the imaginary quadratic field ...
4
votes
1answer
61 views

L-function at s=5 with D=-4?

I want to know the value of $L(5,-4)$. Recall that $$ L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right) n^{-s}. $$ I would like a reference with computations of $L(5,D)$, or more generally, of ...
7
votes
0answers
710 views

How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?

I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
1
vote
1answer
76 views

Unimodular matrices without stable sub-spaces of even weight?

For each N, is there an N×N invertible matrix T over ℤ/2ℤ which does not have a stable subspace of "even weight" -- i.e.  such that there does not exist a set of vectors over ...
70
votes
4answers
2k views

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
8
votes
1answer
147 views

On the set of integer solutions of $x^2+y^2-z^2=-1$.

Let $$ \mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}. $$ The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication. It's known that there is ...
14
votes
4answers
365 views

If $m^2 = (a+1)^3 - a^3$, then $m$ is the sum of two squares.

Prove that: $$\text{If} \space m^2=(a+1)^3-a^3\text{ where}\space m,a\in\mathbb{N} \implies \exists c,d \in\mathbb{N}\space \text{ such that}\space m=c^2+d^2.$$ Maybe it is wrong, if it is let me know ...
5
votes
1answer
110 views

$X^A \equiv B \pmod{2K + 1}$

I recently found this problem which asks you to find an algorithm to find all $X$ such that $X^A \equiv B \pmod{2K + 1}$. Is there something special about the modulus being odd that allows us to ...
2
votes
1answer
584 views

About Self Number, which is found by D. R. Kaprekar.

I'm trying to understand the algorithm to find self-number. But I don't know what does C, k, j, b is mean at this formula. What's that? How do I understand and what should I assign to solve them?
17
votes
5answers
923 views

What interesting open mathematical problems could be solved if we could perform a “supertask” and what couldn't?

If we had a computer that could perform a countably infinite number of steps of a Turing machine, what currently open problems could we solve? I guess a lot of number theory problems could be solved ...
3
votes
3answers
255 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
6
votes
5answers
879 views

Repeating digits in $\pi$

As $\pi$ has infinite digits in its decimal expansion, one could argue that its digits will repeat after a finite number of digits. If so, it is a rational number. What's wrong with this argument?
8
votes
1answer
387 views

if $m^2 = a^3 - b^3$, then $m$ is the sum of two squares.

(Please read "Edit"s and see this.) How could I prove that : $$\text{If} \space m^2=a^3-b^3\text{ where}\space m,a,b\in\mathbb{N} \rightarrow \exists c,d \in\mathbb{N}\space \text{ such that}\space ...
2
votes
1answer
205 views

A question from Titchmarsh's Riemann Zeta Function textbook.

I have one query, concerning the newest edition of this monograph. At page 7, section 1.2, at the bottom of the page, it's written that: " It is easily seen that $\zeta(s)=2$ for $s=\alpha$, where ...
0
votes
2answers
122 views

With which single digit number will these kind of sequences most likely end up with?

If I take any natural number, add the digits together until they produce a one digit number, like in the following example $$5847\ \ \rightarrow\ \ 5+8+4+7=24\ \ \rightarrow\ \ 2+4=6$$ which single ...
4
votes
1answer
489 views

Finding a generator of $(\mathbb Z/p\mathbb{Z})^*$

Is there a method for finding a primitive element (generator) of $(\mathbb Z/p\mathbb{Z})^*$, where $p$ is a prime number?
1
vote
1answer
97 views

Existence of an algebraic integer satisfying a particular condition

Let $K$ be an algebraic extension of the rationals of degree $n$ and let $\{ 1, \omega_2, \omega_3, \dots , \omega_n \}$ be an integral basis for the ring of integers $\mathcal{O}_K$ of $K$. Let ...
1
vote
2answers
1k views

The Binary Representation in Number Theory?

I was reading "Concrete Mathematics" and confronted the Joseph Problem. I'm astonished that the recurrence equation in binary representation is so simple. i.e. for J(n) say J(345), $\text{J(345) = ...