2
votes
1answer
77 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
2
votes
0answers
30 views

Fermat pseudo primes

Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime? In other words, can a Mersenne number be a Fermat ...
4
votes
2answers
110 views

Difficult generating function

Define a beautiful number to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise ...
3
votes
1answer
88 views

Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
2
votes
3answers
103 views

Trouble with inequalities

I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare ...
2
votes
2answers
29 views

Divisibility of sum of squares

I'm currently working through an olympiad problem book that uses the following fact: $3 \mid a^2 + b^2 \implies 3 \mid a$ and $3 \mid b$. I don't see how to show this. I know for that a prime $p$ ...
6
votes
3answers
131 views

Find all real solutions of $6^x+1=8^x-27^{x-1}$

Find all real solutions of $6^x+1=8^x-27^{x-1}$. Things I tried: We want solutions of $$2^x3^x+1 = (2^x)^3-\frac{(3^x)^3}{27}.$$ Write $a=2^x$ and $b=3^x$. This gives $$ab+1 = a^3-\frac{b^3}{27}$$ or ...
2
votes
2answers
39 views

Solve an equation in positive integers

Does $$x^2+y^2=3(z^2+ u^2)$$ have solutions in positive integers? I was assigned this problem, but I am struggling to find a solution. I guess that a proof by contradiction is required.
2
votes
0answers
52 views

Problem on the digits of $n!$

let $m$ be a natural number, is it always possible to find an $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$
2
votes
0answers
93 views

Please help me remembering a problem on $n!$ [closed]

I couple of years ago a friend of mine gave me a problem about the digits of $n!$, I never solved it and I also forgot what the question was. It asked you to prove some fact about the digits of $n!$ ...
4
votes
2answers
117 views

Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$

Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is ...
1
vote
2answers
45 views

Find square roots upto infinte times

Evaluate : $\sqrt{1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}}$ Is it possible to solve in the following way : Let $x=\sqrt{1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}}$ $x^2= 1+ 2 \sqrt{1+3 \sqrt{1+\dots\infty}}$ ...
1
vote
2answers
146 views

Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ consecutive odd integers whose sum is $n^k$

Found these problems in a problem book and got stuck. The book doesn't have solutions to I've come here for help. (1) Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ ...
5
votes
3answers
115 views

Prove that any two numbers of the form $2^{2^n}+1$ are coprime to one another.

Full problem statement: Prove that any two numbers of the follwing sequence are relatively prime: $2 + 1, 2^2+1, 2^4 + 1, 2^8+1, ... 2^{2^n} + 1 $ So far I have tried to use Euclid's algorithm with ...
3
votes
1answer
102 views

Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
1
vote
1answer
181 views

Proving Goldbach's conjecture (hypothetically)

Part $1$. If $\pi(n) \sim \frac{n}{\ln(n)}$ by the prime number theorem, can we treat $\frac{1}{\ln(n)} $ as the probability that a number less than $n$ is a prime number? Say we have some operation ...
0
votes
2answers
91 views

How to show the existence of a number with certain divisibility conditions between two multiples?

How can we show that between two even natural numbers they're exists a natural number that isn't even? How can we show that they're exists a natural number that is odd and not divisible by 3, between ...
0
votes
0answers
29 views

The product of $i$ consecutive natural numbers is divisible for $i!$ [duplicate]

There is a theorem that I've used it a few times, and never saw a demo of it, and when I tried, I could not, commenting with a teacher, it would not give me much attention and said it would use the ...
20
votes
1answer
475 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
0
votes
1answer
103 views

I do not know where to start this challenge, help please

If $x$, $y$ and $z$ are positive numbers such that $1\leq xy+yz+zx\leq3$ which is the set of values ​​of $xyz$? And $x + y + z$?$$$$Knowing that $x, y$ and $z$ $\in\mathbb{R^*}$
1
vote
2answers
86 views

Problem of Ages (Problema das Idades)

English: Somebody help me with this challenge? It's very confusing: Today, both me and my younger brother are between $10$ and $20$ years old. Also, our ages are expressed by prime numbers and the ...
8
votes
2answers
3k views

IMO 2013 Problem 6

Let $n\geq 3$ be an integer, and consider a circle with $n+1$ equally spaced points marked on it. Consider all labelings of these points with the numbers $0,1,\dots, n$ such that each label is used ...
15
votes
4answers
289 views

Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$

Find $x,y,z \in \mathbb Q$ such that: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$$ Here is my thinking: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z\\ \implies \left ( x ...
9
votes
3answers
280 views

show that $a_{n}=\frac{1}{4}[(1+\sqrt{2})^{2n+1}+(1-\sqrt{2})^{2n+1}+2](n>1)$ not have square numbers

Show that the sequence $$a_{n}=\dfrac{1}{4}[(1+\sqrt{2})^{2n+1}+(1-\sqrt{2})^{2n+1}+2]\qquad (n>1)$$ doesn't contain a perfect square. I think this problem is interesting, and my idea: we ...
14
votes
2answers
158 views

Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$

How do I simplify: $$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$ Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
2
votes
1answer
81 views

Imperfect digit-to-digit invariants in Base $10$

$3435 = 3^3 + 4^4 + 3^3 + 5^5$ is an example of a perfect digit-to-digit invariant. Fact: The number of PDDIs is finite for any given base; in particular, for base $10$. Question: Working over base ...
1
vote
0answers
86 views

Find total number of sets of integers which satisfy a given equality and inequality

Compute the total number of different sets of integers a1, a2,..,an which satisfy the following equality and constraints: $$ ...
0
votes
1answer
75 views

A combinatorics problem refer to this problem?

If i define $f(m,n)=$ $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Then prove $$f(m+n,n) - f(m,n) =\frac{n^2-n}{4}$$ for all $m$ and $n$. This question came ...
16
votes
3answers
387 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...
8
votes
5answers
792 views

How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here. How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible ...
13
votes
1answer
306 views

How many $N$ of the form $2^n$ are there such that no digit is a power of $2$?

How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$? For this one the answer given is the $2^{16}$, but how could we prove that that this ...
2
votes
0answers
102 views

Feasibility of a cryptography transformation

This is a follow-up of the question: Transformation We are given $$g^{1/(x+m)},$$ (it is not possible to find $\frac{1}{x+m}$ due to the Discrete log problem), can we find a $k$ such that ...