Questions tagged [number-theory]
Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
35,935
questions
0
votes
0answers
16 views
Rational solutions of quadratic forms
Is there an algorithm or a method that one can use to determine whether an equation of the form $(\text{E})$:
$$ax^2+by^2+cz^2+dt^2=0$$
has a solution $(x,y,z,t)$ in whole numbers. In other words, ...
1
vote
0answers
35 views
Remainder Modulo Question
A number when divided by $902$, $802$ and $702$ leaves remainder of $602$, $502$ and $402$ respectively. What would be the minimum remainder of when the number is divided by $2005$?
My attempt:
For ...
2
votes
0answers
43 views
Special properties of $32$
Is $32$ the only positive integer with the following properties?
It is a Leyland Number, i.e., it is a number expressible in the form $a^b+b^a$ where $a,b>1$ are positive integers.
It is of the ...
6
votes
0answers
38 views
Music and Maths - Is there a way to prove that there are only $7$ Modes of Limited Transposition?
In music, 'Modes of limited transposition' are modes that have a limited availability of transpositions. Unlike a major scale that has $12$ possible unique transpositions, the seven modes of limited ...
-2
votes
0answers
36 views
Number Theory Problem Solving by prove [closed]
If $\gcd(a , 91) = \gcd(n , 91) = 1$, then prove that $91 | (n^{12} - a^{12})$.
I have tried solving this question But I didn't got any solution
1
vote
0answers
30 views
When is $\sum_{n \in \mathbb{Z}} f(n) > \int_{\mathbb{R}} f(x) \, \mathrm{d}x$?
In relation to a research problem, I am facing the problem of showing that
\begin{align}
\sum_{n \in \mathbb{Z}} f(n) > \int_{\mathbb{R}} f(x) \, \mathrm{d}x
\end{align}
where $f$ is a specific non-...
0
votes
0answers
21 views
Show that $\sum_{p\leq x}\log p(\frac 1{\log p}-\frac 1{\log x})=\int ^x_2\sum_{p\leq y}\frac{\log p}{(\log y)^2}\frac {dy}y$.
Show that $$\sum_{p\leq x}\log p\left(\frac 1{\log p}-\frac 1{\log x}\right)=\int ^x_2\sum_{p\leq y}\frac{\log p}{(\log y)^2}\frac {\mathrm{d}y}y$$.
Using $$\int^x_p\frac 1{(\log y)^2}\frac{dy}y=\frac ...
0
votes
0answers
30 views
0
votes
1answer
16 views
Modular form Fourier series calculations
Let $f(z)$ be a modular form of weight $k$ for $\text{SL}_{2}(\mathbb{Z})$ and set $g(z) := f(Nz)$ for some $N>1$, $g$ is a modular form of weight $k$ for $\Gamma_{0}(N)$. I am trying to evaluate $...
22
votes
1answer
181 views
On the parity of $\left\lfloor{\frac{3^n}{2^n}}\right\rfloor$
Let $a_n=(-1)^{\left\lfloor{\frac{3^n}{2^n}}\right\rfloor}$ and $$s_n=\sum_{k=1}^na_k.$$
Is it true that $s_n\le 0$ for all $n\geq 1$ ? (This is true for $n\le 100000$.)
In other words, odd numbers ...
1
vote
2answers
81 views
Simplest proof that $\sum 1/p$ grows like $\log \log x$
I am seeking recommendations for a simple proof that $\sum_{p<x} 1/p$ grows like $\log (\log (x))$.
By simple I mean suitable for secondary school or 1st year university students - not those who ...
1
vote
0answers
22 views
Computing Fourier expansion of modular forms
Say one knows the Fourier expansion $\displaystyle f(\tau) = \sum_{n=0}^{\infty}a_{n}q^{n}$, $q=e^{2\pi i\tau}$, of some modular form $f$ of weight $k$ for $SL_{2}(\mathbb{Z})$.
How does one compute ...
0
votes
1answer
33 views
Why are we able to find the closed form of each number of the form $Ī¶(2n)$, but struggling with numbers of the form $Ī²(2n)$?
It is known that we are able to find the closed form of each number of the form $Ī¶(2n)$, but why are we struggling with numbers of the form $Ī²(2n)$ (Dirichlet Beta function), even though they are very ...
1
vote
1answer
31 views
Counting numbers smaller than $N$ with exactly $k$ *distinct* prime factors
Using common notation, $\omega(n)$ is the number of distinct prime factors on $n$. Similiarly, $\Omega(n)$ is the number of prime factors of $n$, not necessarily distinct: $120=2^{3}\cdot 3 \cdot 5$ , ...
1
vote
0answers
19 views
Irreducible gaussian integers with prime norm in $\mathbb{Z}$ [duplicate]
I need to prove this
Prove that a Gauss integer $z = a +ib$, with $a \ne 0, b \ne 0$ is
irreducible if and only if $a^2 + b^2$ is a prime element in ā¤.
I've already proved that $N(z)$ prime implies, ...
-1
votes
0answers
13 views
A vanishing point of an irreducible representation
Let $X = X_0(1) = SL_2(\mathbb{Z})/ SL_2(\mathbb{R})$, and let $(f_n)$ a cuspidal irreducible representation isomorphic to the principal series.
Could there be some $x\in X$ such that $f_n(x) = 0$ for ...
-1
votes
3answers
47 views
How can I prove that $4^n-1$ is divisible by 3? [duplicate]
I was faced with the hypothesis that $4^n-1$ is always divisible by 3. I believe that this problem can be solved by a proof by induction.
As far as I understand, a proof by induction works by first ...
-1
votes
0answers
14 views
Can I know the cyclic pattern for getting consecutive number arrangements for N=64 whose adjacent numbers when added gives you a square. [closed]
Can I know the cyclic pattern for getting consecutive number arrangements for N=64 whose adjacent numbers when added gives you a square. I tried manually but got struck many a times.
2
votes
0answers
157 views
Relaxation of the $\pi^{\pi^{\pi^{\pi}}}$ problem
My friend showed me this fascinating problem:
Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?.
Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\...
0
votes
1answer
54 views
Irreducible Polynomials and ring of integers [closed]
I came across an exercise in a book which I do not how to solve.
Let $f(x) = x^5 -80x -2.$ Let $\alpha$ be a zero of this polynomial. Furthermore let $K = \mathbb{Q}(\alpha)$ be a number field. These ...
0
votes
0answers
36 views
How do you solve quadratic congruences with unknown modulo using Chinese Remainder Method and Hensel's Lemma
Show that for all positive integers $n$, the following congruence has solutions:
$$(x^2-2)(x^2+7)(x^2+14) \equiv 0 \pmod{n}$$
I need to use the Chinese Remainder Theorem and Henselās Lemma.
So far I ...
4
votes
0answers
77 views
About proving that there are infinitely many prime numbers $p$ such that $\mathrm{ord}_p(a)=\mathrm{ord}_p(b)$
I have seen this question here, however, what interested me the most is the partial answer which uses $Fermat's \ little \ theorem$, but as you can see there, he didn't continue proving that the power ...
0
votes
1answer
35 views
On the density of colossally abundant numbers.
Define $\sigma(n) = \sum_{d\mid n} d$. A integer $n>1$ is said to be colossally abundant (CA) if there exists some exponent $\varepsilon > 0$ such that $$\frac{\sigma(n)}{n^{1+\varepsilon}} \geq ...
-4
votes
2answers
97 views
Can solving the ABC conjecture be casted as a matter of solving some provability or undecidability in FOL PA (Peano arithmetic)?
To the extent that the current alleged proof of the ABC conjecture doesn't seem to have been widely accepted as valid, correct by the mathematics community at large (to say the least), it wouldn't be ...
3
votes
2answers
40 views
Powers of transcendental numbers that lead to integers
For a given real number $x>0$ define $a_k(x)$ as follows for $k \geq 1$:
$a_1(x)=x$, $a_{k+1}(x)=x^{a_k(x)}$, so that $a_2(x)=x^x$, $a_3(x)=x^{x^x}$....
Question 1: Is there a explicit ...
6
votes
0answers
92 views
Generate a random permutation of the first $n$ numbers on the fly
I have a large number of people coming to my shop and want to assign ids from $1$ to at-most $n$ to all of them. While I don't know in advance how many people will arrive, I'll simply stop accepting ...
1
vote
0answers
75 views
Elementary function producing primes
I know there is no known elementary function $f:\mathbb{R}\to\mathbb{R}$ such that $f(n)$ is the $n$-th prime number, but is there a proof for the non-existance of one? How about one which only takes ...
2
votes
0answers
50 views
Interesting sequences when computing nested square roots
In my previous post, I defined
$$f(\color{blue}{m},\color{red}{n})=
\sqrt{2\color{blue}{-}\sqrt{2\color{blue}{-}\cdots\color{blue}{-}
\sqrt{2\color{red}{+}\sqrt{2\color{red}{+}\cdots\color{red}{+}
\...
0
votes
1answer
62 views
How to find to the value of $āx^{p_n}$, when $p$ runs over every odd prime number? [closed]
The original question is:
(1)How to find to the value of $\sum_{p\geq k} \frac{p!}{(p-k)!}x^{p-k}$, where $k$ is fixed, and $p$ runs over every odd prime number that is grater than $k$ and $0<x<...
-1
votes
1answer
54 views
Is a an integer to the power of an irrational number irrational? [closed]
This seemed like a very obvious thing. I was playing around with exponents and logarithms, and I just started wondering, "if $a$$\in$$\mathbb Z$, and $r$$\in$$\mathbb R \setminus \mathbb{Q}$, is $...
4
votes
0answers
36 views
Creating a complete number diagram once and for all
Background
I have been recently searching for a good number diagram I could save for future reference. I was looking for a picture that would show different types or real numbers, and also ...
1
vote
1answer
36 views
What is the relation between order in a number field and order in a quaternion algebra?
The study of number field theory and quaternion algebra theory confuses me a lot:
there are always defined symmetrical quantities (for example the trace, the norm, the orders).
But I can't understand ...
0
votes
2answers
29 views
Understanding a lemma about prime numbers and divisors
In the MCS book from MIT (p. 170) they give the following lemma:
For any prime $p$ and any positive integers $n,x_1, x_2, \ldots, x_n$, if $p \mid x_1\cdot x_2\cdot \ldots \cdot x_n$,
then $p\mid ...
0
votes
0answers
34 views
Formula for the root of $x^2=-3\mod n$ when $n=p_1^{k_1}\cdots p_l^{k_l}$ and $p_i$ primes equal to $1\mod 3$
Consider the equation $$x^2=-3\mod n,$$ where $n=p_1^{k_1}\cdots p_l^{k_l}$ with $p_i$ primes equal to $1\mod 3$. Notice that any such $n$ can written as $a^2+3b^2$ (by a theorem of Fermat) for some ...
0
votes
1answer
62 views
proof check: $f$ maps positive integers to positive integers. $f(m)f(n)=f(mn)$ and $m+n$ divides $f(m)+f(n)$
Find all possible functions $f$ such that $f$ maps positive integers to positive integers, $f(m)f(n)=f(mn)$ and $m+n$ divides $f(m)+f(n)$.
Can we safely say that $f(m)=m^k \cdot a$ (where $a$ is a ...
-1
votes
0answers
35 views
Gauss sums in positive characteristic [closed]
Looking for references on Gauss sums when the multiplicative and additive character take their values in a finite field instead of the complex numbers. Such sums occurred in some recent constructions ...
0
votes
0answers
22 views
Quadratic curves of integers $n=p_1^{k_1}\cdots p_l^{k_l}$ with $p_{i}$ $1\mod 3$ prime
Consider integers of the form $$n=p_1^{k_1}\cdots p_l^{k_l},$$ with $p_{i}$ primes which equal $1\mod 3$.
With help of Mathematica, it seems that there exist quadratic families of subsets of values of ...
0
votes
0answers
23 views
A puzzle problem of Digital VLSI design?
My thoughts:
Here price of chocolate is increasing in Geometric progression with common ratio 2 and first term 1, so on nth day price of chocolate will be $T_{n}=2^{n-1}$ irrespective of wheather ...
1
vote
0answers
103 views
Proof $\frac{(3^n-1)(3^{n-1}-1)…(3^{n-k+1}-1)}{(3^k-1)(3^{k-1}-1)…(3^1-1)}$ is always an integer
Prove that
$$
\frac{(3^n-1)(3^{n-1}-1) \dots (3^{n-k+1}-1)}{(3^k-1)(3^{k-1}-1) \dots (3^1-1)}
$$
is always an integer, for $0\leq k \leq n$ and $n,k \in \mathbb{N}$.
4
votes
1answer
82 views
Understanding the symmetries of f(x)=7x (mod 12)
I found something interesting while studying the circle of fifths.
Define a map $f(x) = 7x$ (mod 12), this models the circle of fifths as a table. The input is the location on the circle of fifths, ...
1
vote
0answers
79 views
To find when $\sqrt{\frac{7^{n}+1}{2}}$ is prime
Find all $n$ such that
$\sqrt{\frac{7^{n}+1}{2}}$ is a prime.
I have tried elementary methods like factorising and using $\pmod{7}$ and so on, but with no luck. I think this question could be ...
6
votes
3answers
184 views
Pascal's triangle curiosity
I noticed the following pattern in the rows of Pascal's triangle:
$$
1 = 11^0\\
11 = 11^1\\
121= 11^2\\
1331= 11^3\\
14641=11^4
$$
at this point I thought maybe this pattern would follow indefinitely, ...
0
votes
0answers
31 views
Problem with PNT proof
I am struggling with some expression involving Mƶbius formula. I try to understand the first equality of the following
For knowledge, in this sum, $S(x,y)$ is defined by
where $P+(n)$ means the ...
-2
votes
0answers
29 views
Representation of Goldbach's Conjecture by removing non-useful odd numbers by different filters. [closed]
Before going on, I would say this post is going to be long and I request you try to read it fully.
Also, this is my first time writing lengthily in LaTex, so please spare me for my naivity. Edit: ...
-1
votes
0answers
31 views
How many times digit 9 appears between 1 to 10000 [closed]
While writing numbers from 1 to 10,000 how many times the digit 9 will be written ?
1
vote
2answers
48 views
prove or disprove in $\mathbb{Z}[i]$ decomposition of real integer to 2 elements with the same norm is unique up to multiplying by unit [closed]
let $a,b,c\in\mathbb{Z}[i]$such that a=bc and say $|b|=B$ and $|c|=C$ . prove or disprove, there is no other decomposition such: $a=de ,|d|=B,|e|=C$
up to multiplying b and c by unity.
Edit: this ...
4
votes
1answer
56 views
Making sure $a$ and $b$ are relatively prime
I came across this interesting problem in the Olympiad Maths challenge practice problem, and it is really fascinating:
Some $n$ numbers are selected randomly from the integers $1$ to $420$. $2$ ...
1
vote
0answers
48 views
When is an elliptic surface non-singular?
I am attempting to understand surfaces of the form $z^2=x^3+Ax+B+Cy+y^3$. I know that an elliptic curve $y^2 = x^3 + Ax + B$ is non-singular iff $4A^3 + 27B^2 \neq 0$. Using this I can see that if $f(...
-1
votes
1answer
17 views
How might one prove, by contradiction, that $n$ is prime if $\mathrm{gcd}((n-k),(n-2k))=1$ for for all $k$ s.t. $1\leq k\leq(n-3)/2$?
How might I go about proving this by contradiction. I came about this on my own, but I know it is known. It's rather obvious if one looks at a binary representation of Euclid's Orchard. Does that ...
0
votes
1answer
25 views
prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$
prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$.
My try: use gaussian integer to claim that we have ufd. the problem, is that k might not be irreducible ...
