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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22
votes
3answers
2k views

Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$

This is a 2015 IMO problem. It seems difficult to solve. Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$. Four such ...
1
vote
0answers
27 views

Erdős' papers on Analytic Number Theőry

My adviser has often mentioned that Paul Erdős' works on Analytic Number Theory contain a myriad of techniques that any number theorist must know. What are some of his papers in Analytic Number Theory ...
3
votes
1answer
45 views

How shall I calculate $\sum\limits_{d\nmid n}\mu(d)$

Today when I was studying Apostol's Analytical Number theory, I came to know about the formula $\sum\limits_{d|n}\mu(d)=1$ if $n=1$ and $0$ otherwise. I understood the technique and then using the ...
4
votes
2answers
47 views

Closed form expression for products

How can I find a closed form expression for products of the following form $$\prod_{k=1}^n (ak^2+bk+c)\space \text{?}$$
0
votes
3answers
67 views

Calculate the exact value of the following expression [closed]

I propose the following exercise. Calculate the exact value of$$P=\dfrac{(10^{4}+324)(22^{4}+324)(34^{4}+324)(46^{4}+324)(58^{4}+324)}{(4^{4}+324)(16^{4}+324)(28^{4}+324)(40^{4}+324)(52^{4}+324)}$$ ...
3
votes
2answers
34 views

Is it known whether there are ever infinitely many primes of the form $\prod_i p_i^{n_i} + 1$ where the $p_i$ are fixed primes but the $n_i$ can vary?

So if we fix finitely many primes $p_i$, where one $p_i$ is $2$, but let the powers $n_i$ on the $p_i$ vary, is it known whether it is ever possible to have infinitely many primes of the form $\prod_i ...
3
votes
1answer
78 views

Infinite number of primes of the form $2^x \cdot 3^y + 1$?

Are there an infinite number of primes of the form $2^x \cdot 3^y + 1$? I really have no idea where to start with this. I thought of it because it would imply an affirmative answer to this recent ...
4
votes
1answer
75 views

Does Euler's $\phi$ function have the same value in arbitrarily large subsets of $\mathbb{N}$?

As my most recent question still does not have any answers and it appears to be a difficult problem, I propose the following problem (that seems easier), but which I still could not manage to solve: ...
0
votes
0answers
18 views

Reference on the normality of some numbers

I'm searching for a contemporary reference on the normality on basis 10 of the Champernowne and Erdös-Copeland constants, defined as $$ C=0,12345678910111213... $$ $$ E=0,23571113...$$ that is, the ...
2
votes
1answer
48 views

Comparing $\pi(x)$ and $\pi^{(k)}(x)$

We say a k-almost prime is an integer that results as the product of k prime, counting repetition. For example, $12$ is a $3$-almost prime as $12= 3 \times 2 \times 2$. Additionally, we define ...
2
votes
1answer
66 views

7, 13, and 37 will always divide numbers such as 151515, 474747, 868686

I happened to be looking at two digit numbers that repeated 3 times, such as 151515, 474747, 868686, etc., 3 always goes into these, which is self explanatory because the total of the numbers will ...
0
votes
0answers
9 views

Convenient goedel numbering for finite register machines?

Overview I'm trying to find a goedel numbering for finite register machines, which is convenient in two ways: when ordering machines by their numbering, simple machines shall come first, i.e. ...
1
vote
1answer
81 views

interesting remainder problem

Let $A=\left( 2\sin{\dfrac{7\pi}{18}}+1 \right)^{2556}$. How can one find the remainder from the division of $\lfloor A \rfloor$ by $17$ ? I have no idea. Thank you.
2
votes
1answer
71 views

How do computers generate primes so quickly?

From what I understand, when a computer encrypts a file using an encryption standard like RSA, one of the steps is to generate two large primes, and multiply them together. I have created RSA keys on ...
5
votes
2answers
530 views

Is this extension of Goldbach's conjecture obviously false?

Goldbach's conjecture is: Every even integer greater than $2$ can be expressed as the sum of two primes. Extension of Goldbach's conjecture is: Every number from $p\mathbb{Z}$ greater than ...
3
votes
0answers
57 views

Is the set of integers so that $n!+1$ divides $(2012n)!$ finite or infinite?

I am having trouble with this problem. We have to determine whether the set of integers such that $n!+1$ divides $(2012n)!$ is finite or infinite. Basically we have to determine if the prime factors ...
1
vote
1answer
40 views

Cohomology of Severi-Brauer varieties

What can be said about Galois-module structure of $l$-adic cohomology of a Severi-Brauer variety over a local field? In particular, I'm interested in the proof of the proposition given at the top of ...
1
vote
3answers
54 views

Find the number $ccbb$

If the number has $4$ digit $ccbb$ and it's full square, then find that number. I have tried and I got $88^2=7744$ but my way has no prove for it, if any one have away, I'll appreciate it.
-3
votes
3answers
65 views

Find the value of $x$ [closed]

Find the value of $x$, $$\left ( \frac{4}{\sqrt{3}-\sqrt{2}} \right )^{4-x}=\left ( 80+32\sqrt{6} \right )^{x}$$ any help?
2
votes
1answer
30 views

System of Congruences with Special Symmetry

Show that the following system of congruences \begin{align} \begin{cases} 3 x^4 - 7 x^2 y^2 - 7 x^2 z^2 - 35 y^2 z^2 \equiv 0 \pmod{p} \\ 3 y^4 - 7 x^2 y^2 - 7 y^2 z^2 - 35 x^2 z^2 \equiv 0 \pmod{p} ...
3
votes
0answers
50 views

Cryptosystem ElGamal

If $p$ is an odd prime and $n$ natural,it is known that the group $Z^*_{p^n}$ is cyclic.Explain why the selection-choice of the group $Z^*_{{3^{1000}}}$ for the construction of a cryptosystem ElGamal ...
1
vote
3answers
55 views

Prove this function is injective $f(x)=x+\mod(x,7)$

Prove this function is injective $f(x)=x+\mod(x,7)$. Attempt: I tried separating in two cases: $x \equiv y \pmod 7$ and $x \not \equiv y \pmod 7 $: First case: $$f(x)=f(y) \iff x+\mod(x,7)=y+ \mod ...
0
votes
2answers
38 views

Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$

I am writing a program to output all $a,b \in \mathbb{N}$, where $a \le b \le n$ (for a given $n \in \mathbb{N}$), such that $$ \frac{ab}{2\sqrt{ab}+a+b}=c\in \mathbb{N} $$ For example, $a=9$, ...
0
votes
0answers
14 views

On finiteness of solutions to Catalan/Pillai diophantine equation

Give a fixed positive integer $k$. Is there a constant $c(k)$ depending only on $k$ such that the number of solutions to the equation $$ ...
0
votes
1answer
23 views

Real Primitive Characters and Quadratic Forms

since I do not know algebra very well, I have some troubles with the following concepts: I want to ask something about quadratic fields and quadratic forms. Actually my main question is the ...
6
votes
2answers
311 views

Smallest Positive Integer Not Coprime to a Collection of Consecutive Integers

Let $n\in\mathbb{N}$. Define $f(n)$ to be the smallest positive integer $m$ such that there exists a positive integer $k$ for which $k+i$ is not relatively prime to $m$ for every ...
1
vote
4answers
96 views

Find the value of $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$

If $$x+y+z=7$$ and $$\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=\frac{7}{10}$$ Find the value of $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}$$ I tried but I got nothing
6
votes
0answers
134 views

Is there a quicker way to generate integers which are hard to factor than multiplying two large primes?

An easy way to generate an integer which is hard to factor is to find two large primes and multiply them. As a bonus, you know the factors. I'm interested in whether it's possible to find integers ...
4
votes
1answer
55 views

Discrete Fourier Transform of $\omega^{n(n-1)/2}$

For the sequence $x_0$, $x_1$, $\ldots$,$x_{N-1}$, let $\omega=e^{2\pi i/N}$ and define the discrete Fourier transform as $$X_k = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x_n\omega^{nk}\,.$$ I'm interested ...
7
votes
0answers
127 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
1
vote
0answers
43 views

Is this Riemann Zeta function integral formula known about?

I discovered that $$\zeta(s)=\int_0^1\frac{(-\log(1-x))^{s-1}}{x(s-1)!}dx.$$ Is this an obvious result that is not worth much interest or is this new and unique?
2
votes
0answers
38 views

Errata for Rosen and Ireland's “A Classical Introduction to Modern Number Theory”?

Anyone know of an errata list for Rosen and Ireland's "A Classical Introduction to Modern Number Theory"? I checked on Springer's web site, but did not find anything.
0
votes
1answer
36 views

Simple Linear Diophantine Equation - problem with proof.

I'm reading up on diophantine equations and one of the theorems is that "if $x,y$ is any solution of $ax + by = c$, then it is of the form $x_0 +\dfrac{b}{d}t ,\, y_0 - \dfrac{a}{d}t$ where $d = ...
1
vote
2answers
21 views

Distribution of decimal repunit primes

The prime number theorem describes the distribution of prime numbers in positive integers. Is there a similar theorem describing the distribution of primes among positive integers of the form ...
-1
votes
4answers
44 views

Prove $P+G$ and $(P-1)!x-(G-2)$ are coprime

I've been interested in studying twin primes--- not to try to prove the nearly unprovable twin primes conjecture, but just in and of themselves, with the hope that maybe I could find something that ...
0
votes
0answers
20 views

Multivariate “base polynomials”

Fix a positive integer $b$. For any positive integer $N$ whose base-$b$ expansion is $\sum \alpha_k b^k$, define $p_N(x)$ to be the polynomial $\sum\alpha_k x^k$. Goal: Produce a polynomial ...
0
votes
1answer
54 views

Prove that if a|b and b|a, then a=b or a=-b

I'm having difficulty proving that if a|b and b|a, then a=b or a=-b. Logically, it makes sense to me, but I don't know how to express it.
0
votes
1answer
41 views

Roots of polynomials $\pmod {p^2}$

Let $f(x)=x^l+x^{l-1}+x^{l-2}+\cdots+x^2+x+1$, $p$ being prime, $f(x)\equiv 0\pmod {p^2}$ if $p\equiv 1\pmod l$ or $p=l$ has $l$ roots, otherwise it has none. So what can be said about ...
1
vote
0answers
27 views

Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...
5
votes
0answers
34 views

Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
3
votes
1answer
71 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
-2
votes
1answer
76 views

Can anyone Find the error below [duplicate]

If: $$S=1+2+4+8+....+2^n +...$$ So we get $$2S=2+4+8+...$$ $$2S+1=1+2+4+8+16...$$ $$2S+1=S$$ $$2S-S=-1$$ $$S=-1$$ Is there error, and if there's, why? I want athletic explanation.
2
votes
1answer
49 views

Sum of permutations of a number

If I have a number (without any 0's) such as 1112334, how would I sum the permutations of its digits (excluding duplicates)? I am assuming there is a closed form involving factorials or combinatorics. ...
0
votes
2answers
54 views

Find integer solution of sysem of quadratic equations [closed]

If: $a,b,c$ positive integers, where $a\geq b\geq c$. such that: $$a^2 - b^2 - c^2 +ab=2011$$ $$a^2 +3b^2 +3c^2 -3ab-2ac-2bc=-1997.$$ Find the value of $a$ I tried, but I got nothing. Source: 2012 ...
1
vote
0answers
43 views

Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
7
votes
1answer
52 views

Does there exist a prime number $p$ such that $p\mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal?

Problem: Prove or disprove: there exists a prime number $p$ such that $p\mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal, where $\mathcal{O}_K$ denotes the ring of algebraic ...
1
vote
1answer
46 views

Finding remainder of negative number

Recently my colleague ask one mathematical question which is, What is the quotient and remainder of $(-29)/7$? and my answer was that quotient is $4$ and remainder is $-1$ and he told me I'm ...
-1
votes
0answers
33 views

How could one invert this summation?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
2
votes
1answer
47 views

Inequality of logarithm of the tail of the Euler product

I would want to know if this inequality holds: Let $x$ be a positive integer and $b>1$ be a real, then $$ \sum_{p> x}\log(1-p^{-b})\ge-\frac{b}{x}, $$ where the sum is over all prime $p\ge x$. ...
2
votes
2answers
42 views

Variation of Division Algorithm

How to derive this version of the division algorithm . For integers a, b with b ≠ 0 there exist unique integers q and r that satisfy a = qb + r, where -1/2|b| < r ≤ 1/2 b. I started off with ...