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 algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
40,962
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$a_n=2^n+3^n+6^n-1$. Find all positive integers that are primes to all terms of the sequence.
Let the sequence $a_n=2^n+3^n+6^n-1, n\in\mathbb N_{> 0}$. Find all positive integers that are prime to all terms of this sequence. I have no idea how to approach this, but I know that I CAN'T use ...
- 4,930
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3
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It is possible to show square as sum n distinc squares, where n > 2?
We know that $5^2 = 3^2 + 4^2$.
But it is possible to show $a^2$ as the sum of three or more distinct squares?
Something like that exist?
$a^2 = b^2 + c^2 + d^2$, where $b$, $c$, $d$ are distinct ...
2
votes
2
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222
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How find this minimum of the $q$, if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$
let $p,q$ is postive integer,and such
$$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$
Find the minimum of the $q$
maybe can use
$$95q>36p$$
and $$37p>96q$$
and then find this minimum of ...
- 93.4k
2
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3
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remainder when 67896789...(300 digits) divided by 999
What is the remainder when 678967896789... (300 digits)is divided by 999?
i tried to divide it manually to find some pattern in remainder. But was getting bit lengthy.
so please suggest me some short ...
2
votes
2
answers
401
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Integer solutions of the equation: $x^2+y^2+z^2=kxyz$
Given the equation:
$$x^2+y^2+z^2=kxyz$$ with:
$(k,x,y,z)\in\mathbb{N}$,
the only solution for $k=2$ is:
$x=0,y=0,z=0$.
For what values of $k$ the equations has solutions in which $x,y,z$ are ...
- 10.6k
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3
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142
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Trying to prove that there are no p and q such that $|\sqrt5 - p/q| < 1/(7q^2)$.
Like the title says, I'm having trouble proving that there are no integers p and q such that
$|\sqrt5 - p/q| < 1/(7q^2)$. I was given the hint that $|(q\sqrt5 - p)(q\sqrt5 + p)| \geq 1$, but I don'...
- 575
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1
answer
676
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What are some of the more efficient ways of studying for an Olympiad?
This September I am participating in a competition called the Australian Intermediate Mathematics olympiad, and you may not have heard of it but it's very similar to the AIME. Could you please tell me ...
anon
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2
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285
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What does $conclude$ mean in this sentence?
My friend asked me a question, but I don't know the meaning of the sentence Conclude that $\sigma_n$ is a ring automorphism here, does it mean Prove that $\sigma_n$ is a ring automorphism or Make the ...
- 15.8k
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Is this a true theorem? [closed]
I'm trying to prove the existence of the following theorem:
If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$
Is this theorem true? I think it is, but I don't know how to prove it!
Thanks!
- 21
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2
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90
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Find the remainder of a division
Which is the remainder of the division $985^{423}:98$?
That's what I have tried so far:
Let $a=985,n=98$. Then $(a,n)=1$ and $\varphi(n)=42$.
So, we have that $985^{42}\equiv 1 \pmod{98}$. Hence,
$$\...
- 7,833
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1
answer
212
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What is the smallest number k, such that $k^{2014}+2014$ is prime?
What is the smallest number k, such that
$$k^{2014}+2014$$
is prime ? I checked upto $k= 24000$ and did not find a prime.
Since the numbers do not grow very fast ($k=92204$ produces a $10 000$-...
- 84.6k
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2
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185
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Let $p$ be the largest prime less than or equal to $n$. Is $n$ $\underline{<}$ $p^2$
Fix $n$ to be some positive integer greater than or equal to 2. Let $p$ be the largest prime less than or equal to $n$. Is $n$ $\underline{<}$ $p^2$?
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Does there exist an infinite sequence $p_0,p_1,p_2...$ of prime numbers such that $p_k=4 p_{k-1}\pm 1$
$k \in Z^+$
firstly we know that there exists infinetly many primes of the form $4n+1$ by FTA
also we see that if we consider finite primes say to $n$ then the recursive formular can be expressed as:...
- 1,032
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1
answer
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Proof on a conjecture involving $d(N)$
Let $d(n)$ denote the number of divisors of $n $.
Then let $N$ be a number such that $d(N)$ divides $N$ . Also let $I= \frac{N}{d(N)}$ which is defined as the "Index of Beauty of $N$ ".
Then prove ...
- 4,012
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2
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477
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AIME number theory problem (unique factorization domains)
I'd greatly appreciate some help with the following problem, from a mock AIME I took.
Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
- 2,640
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$n! =$ the product of consecutive integers. [duplicate]
Can $n!$ be the product of $k$ consecutive integers for $k > 1$? (Not including the degenerate cases such as when $k = 2$, then $1\cdot2 = 2!$ and $2\cdot 3 = 3!$, and so on.)
I am asking not for $...
- 19.6k
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1
answer
136
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Between Mertens' theorems
It is well-known that
$$
\sum_{p\le x}\frac{\log p}{p}=\log x+O(1)
$$
and
$$
\sum_{p\le x}\frac1p=\log\log x+M+o(1).
$$
What is the order of
$$
\sum_{p\le x}\frac{\sqrt{\log p}}{p}
$$
?
- 32.1k
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2
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857
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Divisibility by 8 when converted in base 10
How many 7-digit numbers are there in binary(base-2)? How many of them are divisible by 8 when converted to base 10?
My approach:
In binary system, only two digits are there i.e. 0 and 1. So except ...
- 179
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1
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Limit of $\sum\frac{1}{p(\pi(n))}$
Let $p(n)$ be the nth prime and $\pi(n)$ the number of primes not exceeding n.
I wonder if we can show that
$$\tag{1} S = \sum_{n= 2}^k \frac{1}{ p (\pi (n))} \sim \log k. $$
We know by comparison ...
- 10.2k
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3
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Solving $ x^3 - 3x + \sqrt 2 = 0$
First of all, I'm going to use plain text for my formulas since I'm using a screen reading program, which makes it very difficult to read the highly non-standard format commonly used here, sorry about ...
- 31
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votes
2
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752
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Possible remainders when $3^{4n-2}+ 2^{6n-3} + 1$ is divied by $17$
The numbers in the form of $3^{4n-2}+ 2^{6n-3} + 1$, where $n$ is a positive integer, when divided by $17$, has possible remainders?
- 345
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3
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How to find all the primitive roots in $\mathbb{Z}/49\mathbb{Z}$.
I need to find all the primitive roots of 49. First note, $ ϕ(49) = 42 $
Is there an easier way to go about trying all numbers less than $42$ to find the primitive roots of $49$ if we already know ...
- 35
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2
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212
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Quadratic Forms and Congruences
How does one prove (the non-trivial direction) that, for $n \in \mathbb{N}$,
$x^2 + y^2 + z^2 = n$ solvable $\iff$ $x^2 + y^2 + z^2 \equiv n\ (m)$ solvable for all $m$?
In particular, is there a ...
- 534
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How to factor 30 digit number
I need to find the prime factorization of a number having 30 digits.
I used the Pollard rho method but unfortunately it is not sufficient enough.
It needs a more advanced prime factorization process....
- 357
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votes
3
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144
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Whether twin primes satisfy this one?
It seems that difference of squares of any twin primes $+1$ will always lead to
number which might be
a) A square of a twin prime
b) Itself a twin prime
$C$ = ($A^2$-$B^2$ )+$1$ ------> $(1)$
Where ...
- 597
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votes
2
answers
94
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Closed form way to express $a^1 + a^2 + a^3 + a^4 \cdots a^n$, given $a$ and $n$?
Is there a way to sum up consecutive powers like this when we know the value of $a$ using a closed form expression or do we actually have to add up each one?
By closed form I mean in the same way ...
- 923
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Whether any even number can be written as sum of odd no of primes?
Whether any even number can be written as sum of odd number of primes?(3,5,7.. primes) I know that Goldbach's conjecture state that a even number can be written as sum of two primes.
D=A+B+C+...+n ...
- 597
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Find prime numbers $a, b, c$ such that $a^b+c$ is a prime
Please help me find all $a, b, c \in \mathbb{P}$ such that $a^b+c$ is a prime
Example I just can find: $2^3 + 5 = 13 \in \mathbb{P}$
- 2,967
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continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$
Hensel's lemma implies that $\sqrt{2}\in\mathbb{Q_7}$. Find a continued
fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$
- 539
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4
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Exercise on Fundamentals of Divisibility: Factorization Domains
Set $R=\mathbb{Z}[\sqrt{10}]$. Show that in $R$ every element $\alpha\not=0$ is a product of irreducible elements, but $R$ is not a unique factorization domain.
I have shown that $R$ is not a unique ...
- 421
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97
views
numbers of subsets for a set $A$ for which the equation $x+y=2n+1$ hasn't solutions.
Find the numbers of subsets for the set: $\displaystyle A= \{1,2,\ldots,2n\}$ for which the equation $\displaystyle x+y=2n+1$ has not solutions.
I have no idea. Thanks for your help.
- 6,790
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Is $x^y$ - $a^b$ divisible by $z$, where $y$ is large?
The exact problem I'm looking at is:
Is $4^{1536} - 9^{4824}$ divisible by $35$?
But in general, how do you determine divisibility if the exponents are large?
- 433
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3
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Is 2 a quadratic residue modulo $r=\frac{p^m+1}{2}$?
Suppose that $$r=\frac{p^m+1}{2}$$ is a prime number, where $p$ is also prime. Does the equation $$p^{2}-2\equiv 0 \pmod{r}$$ have any solutions?
Thanks in advance.
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Is $n = k \cdot p^2 + 1$ necessarily prime if $2^k \not\equiv 1 \pmod{n}$ and $2^{n-1} \equiv 1 \pmod{n}$?
$p$ is an odd prime and $k$ is a positive integer.
Let $n=k \cdot p^2+1$.
If $2^k \not\equiv 1 \pmod n$ and $2^{n-1} \equiv 1 \pmod n$, is $n$ prime? If it is, why?
- 57
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Legendre Symbol Problem
I am doing revision for my number theory exam and I am stuck on the following question.
Let $x$ be an even integer. Show that every prime divisor $p$ of $x^4 + 1$ satisfies $\big(\frac{-1}{p}\big)$ = ...
- 1,359
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Proof that $ (a)^n \bmod n^2 = (a \bmod n)^n \bmod n^2$
Proof that $ (a)^n \bmod n^2 = (a \bmod n)^n \bmod n^2$
I did try a couple of examples and they do seem to work, but I just can't get why it works.
- 560
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2
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397
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Heuristic Proof of Hardy-Littlewood Conjecture for 3-term Arithmetic Progressions
The Hardy-Littlewood Conjecture for 3-term arithmetic progressions is that
$$
\# \{ x,d \in \{1,\ldots,N\} \, | \, x,x+d,x+2d \text{ are all prime} \} \sim
\frac{3}{2} \prod_{p > 2} \left(1+\...
- 61
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votes
2
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If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
How to prove that:
If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
This statement is generalization of the statement from my previous question.
I have checked for many $(a,b)$ pairs ...
- 12.9k
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3
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Proof: if $n=pq$ then $p-1\mid q-1$ and $q-1\mid p-1$
Now I'm asking my first question to understand a specific proof:
Let $n=pq$ and $q,p \in \mathbb{P}$. Then we get $p-1\mid n-1$ and $q-1\mid n-1$, because there are prime integers mod $p$ and mod $q$....
- 3,381
2
votes
2
answers
114
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Modular functions and modular forms
A modular form of weight $k$ as a holomorphic function satisfying $f(\gamma z)=(cz+d)^kf(z)$ (automorphy relation of weight $k$) on the upper-half plane and also holomorphic at cusps.
A modular ...
- 1,373
2
votes
1
answer
172
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Missing pattern in solvable negative Pell equation
Considering the negative Pell equation $ x^2 - Dy^2 = -1 $, I know that a necessary condition for solvability is that
$ D = a^2 + b^2$, with $ a,b $ positive integers.
If I fix $ b = 1 $, NPE is ...
- 743
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votes
3
answers
54
views
Order of Odd Elements in $\mathbb{Z}_{2^n}$
I'm wondering if there is a way to calculate the order of odd numbers in the cyclic ring $\mathbb{Z}_{2^n}$. I found a paper that shows that for any odd $x$, $x^{2^{n-2}} \equiv 1 \mod{2^n}$, and that ...
2
votes
2
answers
120
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Writing a square as a sum of three non-zero squares in geometric progression
Let $k$ be a given positive integer. I want to solve the following system of Diophantine equations: $$\begin{cases} a^2 + b^2 + c^2 = k^2 \\ b^2 = ac \end{cases}$$
where $a, b, c \in \mathbb{N}$ are ...
- 2,619
2
votes
1
answer
117
views
Relating the two errors terms in the prime number theorem
I would like to show that the two errors terms in the prime number theorem $\pi(x)-\frac{x}{\log{x}}$ and $\psi(x)-x$ are quite similar (or differing by something like a factor of $(\log{x})^{\...
- 474
2
votes
4
answers
158
views
Find the mystery fraction
Question: There is a fraction with integer numerator and integer denominator, each smaller than $100$, whose decimal expansion begins with $0.11235\ldots$. Both the numerator and the denominator are ...
- 31
2
votes
1
answer
79
views
Find three different number $m,n, k\in\mathbb{N}$ with $m, n, k\geq 3$ such that $mnk=2(mn+nk+mk)$
In my research, I need to solve the equation $mnk=2(mn+nk+mk)$ in natural number set, such that $n, m, k\in\mathbb{N}$ are different number and $n, m, k\geq 3$.
I know that $(n, m, k)\in\{ (3, 7, 42)...
- 1,287
2
votes
1
answer
328
views
What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$?
What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$, where $p$ is a prime number and $p>3$?.
Calculations suggest that the number of solutions to this equation is $p$ if $p\...
user998254
2
votes
2
answers
93
views
Find $\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$
Define $f(n)=\sqrt[2]{n^4+\frac{1}{4}}-n^2.$
Find $$\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$$
I tried to simply $f(n).$ So rationalising, we get $$\sqrt{\frac{1}{2}-f(n)}\sqrt{\frac{1}{...
- 1,607
2
votes
1
answer
143
views
Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$
I'm investigating the behavior of the following function as $x\to \infty$:
$$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$
where $J_k(n)$ is the Jordan's totient function
$$J_k(n):=n^k\prod_{p\mid n}(1-...
- 5,187
2
votes
3
answers
304
views
Find all pairs of primes $(p, q)$ such that $p^2 + 6pq + q$ is a perfect square [closed]
Find all pairs of primes $(p, q)$ such that $p^2 + 6pq + q$ is a perfect square.
This is a problem I encountered a month ago. I remember having had multiple attempts, but they didn't seem to lead ...
