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)

4
votes
0answers
116 views

Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...
4
votes
0answers
59 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
4
votes
0answers
101 views

How find this equation all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$ My try: If $n>1$ is odd, then note $$\left(n^{\frac{n+1}{2}}\right)^2<n^{n+1}+n-1<\left(n^{\frac{n+1}{2}}+1\right)^2$$ so ...
4
votes
0answers
140 views

Algorithm for comparing the size of extremely large numbers

Is there a simple algorithm to decide which of the numbers $$a \uparrow ^b c \text{ and } d \uparrow ^e f$$ is the bigger one ? Using the hyperoperation, the numbers can be denoted with ...
4
votes
0answers
58 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
4
votes
0answers
49 views

List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
4
votes
0answers
120 views

References request: Ramanujan's tau function.

References request: Ramanujan's tau function. Let $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n$, $q=e^{2\pi i z}$. How can one show the following using representation ...
4
votes
0answers
70 views

Intersection between OEIS-A050808 and OEIS-A080053

Is there any result on the intersection between A050808 and A080053: OEIS-A080053 = 1, 2, 4, 5, 6, 10,... (Exp(n) is further from an integer than any previous exp(k)) OEIS-A050808 = 1, 2, 18, ...
4
votes
0answers
112 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
4
votes
0answers
66 views

Is there always a prime number in the form of $4k+1$ between $[n, 2n]$ for every large enough $n$?

Is there always a prime number in the form of $4k+1$ between $[n, 2n]$ for every large enough $n$? I guess it is known as a classical result. Is there any reference for it? Thanks!
4
votes
0answers
69 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
4
votes
0answers
66 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
4
votes
0answers
416 views

Totient summatory function

Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a ...
4
votes
0answers
103 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
4
votes
0answers
122 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
4
votes
0answers
177 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
4
votes
0answers
89 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
4
votes
0answers
123 views

Adelic lattices

Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
4
votes
0answers
134 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
4
votes
0answers
215 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
4
votes
0answers
407 views

The greatest prime factor of $6n+1$

Let $p(n)$ be the greatest prime factor of $n$. Denote $a_1(n)=p(6n+1),a_{k+1}(n)=p(6a_k(n)+1)).$ Is it true that for $\forall n\in \mathbb N,$ $\exists c,t \in \mathbb N^{+}:k>c\implies ...
4
votes
0answers
87 views

Surjective map in Galois cohomology?

Let $p$ be a prime number, $\mathbb{F}_p$ the field with $p$ element and $\omega$ the mod $p$ cyclotomic character. Let $K$ be a finite extension of $\mathbb{Q}_p$ (the field of $p$-adic numbers) and ...
4
votes
0answers
159 views

Solve in integers $ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$

Solve in integers: $$ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$$ My idea: $$\Longleftrightarrow (y^3+xy-1)(x^2+x-y)-(x^3-xy+1)(y^2+x-y)=0$$ $$\Longleftrightarrow ...
4
votes
0answers
88 views

Linear independence of $\cos(n\theta)$

I was trying to see if the cosines of the (certain) integer multiples of a certain angle were linearly independent over $\mathbf{Q}$. In particular I was looking at when $\theta = ...
4
votes
0answers
90 views

Bounds for multi-dimensional Kloosterman Sums

I'm looking for a general bound (in terms of $p$) for the Kloosterman sum, working in $\mathbb{F}_{p}$, $$\sum\limits_{x_{1} \dots \ x_{n} = a} \psi(x_{1} + \dots + x_{n})$$ for $\psi$ a nontrivial ...
4
votes
0answers
191 views

possible prime factors of $4^{444}+3$

I have not factored the number $4^{444}+3$ yet. I wonder, though, if there are restrictions for possible prime factors p. The only obvious restriction is, that -3 must be a quadratic residue of p. ...
4
votes
0answers
264 views

Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is coprime to $10$,such that the period of the decimal expansion of $1/n$ is a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If $n-1=2^xc$ ...
4
votes
0answers
77 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
4
votes
0answers
106 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
4
votes
0answers
113 views

Characterization of quadratic polynomials over $\mathbb{Z}/p\mathbb{Z}$

This is a nice question which I'd like to share with everyone. Let $f:\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$ be a function s.t. for each $a \in (\mathbb{Z}/p\mathbb{Z})^{\times},$ the ...
4
votes
0answers
155 views

Primes dividing sequence

Define a sequence $a_n$ as such. $a_0=1$, $a_n=a_{n-1}+a\lfloor\frac{n}{3}\rfloor, \forall n\ge1$ Find all primes $p$ such that p divides infinitely many values of $a_i$. Edit: This is extension to ...
4
votes
0answers
365 views

Fermat Last Theorem for 13,17,19

I am asked to prove the Case I of Fermat’s Last Theorem for the exponents 13, 17 and 19. FLT states that the equation $x^n+y^n=z^n$ has no integer solutions for and $x,y,z$ different from zero. ...
4
votes
0answers
457 views

Defining the Riemann-Roch space of a divisor

I'm doing a course on elliptic curves. It starts with a bit of a crash course in algebraic geometry, giving statements alone. We were given the following definition The Riemann-Roch space of $D$ ...
4
votes
0answers
49 views

S-Unit equations in cyclotomic fields

By a Siegel's result, one knows that there exist only finitely many solutions of the equation: $$x+y=1$$ where the unknowns $x$ and $y$ are units in the ring of integers of a cyclotomic field. Do you ...
4
votes
0answers
121 views

Solutions to simultaneous excluded congruences

I'm interested in the smallest solution to a family of "excluded congruences." To be precise, let $p_1 < \ldots< p_k$ be a sequence of primes and consider the constraints $$ x \not\equiv a_1 ...
4
votes
0answers
165 views

Does Hensel prove LTE?

Can the Lifting The Exponent Lemma ( found at http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf) be proven with Hensel's Lemma? How?
4
votes
0answers
304 views

Smooth numbers algorithm

I am trying to understand quadratic sieve algorithm and now I am thinking of the way to check if number is smooth over a factor base? For example, say I have number $n = 87463$. First,I find bound $B ...
4
votes
0answers
122 views

Number Theoretic Game

2 players A and B play a game. At the start of the game, $n$ positive integers (not necessarily distinct) are written on a notebook. First, player A chooses a number from the notebook and declares it ...
4
votes
0answers
526 views

Obtain a contradiction

Motivation : The motivation is to show that the equation $x^{2b}.x^{2a} +(3-x^{2b}) x^{a} + (1-s^2)=0 $ has no solutions in integers for any values of $x,b,a,s$ ( choosen as per the constraints ...
4
votes
0answers
102 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
4
votes
0answers
137 views

Liftings in unramified extensions of $\Bbb Z_p$

[Edit : I have changed the formulation of the question. Sorry for the trouble] Here is a stupid question, maybe trivial. Let $p$ be a prime number, $q = p^n$ where $n$ is an integer, $R = ...
4
votes
0answers
417 views

sum of $n^{th}$ powers of prime factors of $x$

Starting with a positive integer $x$, find the sum of the $n^{th}$ powers of the prime factors of $x$, including multiplicities. Then find the sum of the $n^{th}$ prime factors of the result etc. ...
4
votes
0answers
757 views

Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
4
votes
0answers
150 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
4
votes
0answers
318 views

Sum of the primitive roots

It is well known that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \bmod{p}$. But I can't see why this result is interesting or useful. Can someone please enlighten me?
4
votes
0answers
137 views

How to prove that polynomials with integer coefficients generically have full Galois group

Based on the graphic in the MathWorld article on the quintic equation it seems very likely that following statement is either true or trivially false in a way that can be easily remedied by adding an ...
4
votes
0answers
147 views

Champernowne-like squares, are there any?

I read about the Champernowne constant on Wikipedia a couple of days ago, and I got curious about something similar: is there some "Champernowne-like" number; that is, a concatenation of all numbers ...
4
votes
0answers
139 views

Eigenvalues of the $p$-adic Harmonic oscillator?

Given a prime $q$, what are the values of the $p$-adic Harmonic oscillator that is the solution to the following $p$-adic differential equation? $$ -D^2_q f(x)+ x_q^2 f(x) = E_n f(x) .$$ What are ...
4
votes
0answers
47 views

When does PSL(5,q) have order exactly divisible by a specific odd power of 5?

In a misguided attempt at answering a question on divisibility of simple group orders, I looked at $\newcommand{\PSL}{\operatorname{PSL}}v_p(|\PSL(p,q)|)$ which went smooth enough for $p=2$ and $p=3$, ...
4
votes
0answers
65 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?