Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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17
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6answers
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Visual explanation of the following statement:

Can somebody fill me in on a visual explanation for the following: If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$ I know ...
17
votes
3answers
4k views

Largest prime number with all digits different

What is the largest prime with distinct digits? (It is certainly less than ten digits long.Can you explain it why?
17
votes
5answers
1k views

Intervals that are free of primes

How can I prove that exists intervals as large as I want that are free of primes? I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.
17
votes
9answers
4k views

Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer

Please help me with solving this : prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, ...
17
votes
5answers
22k views

Is there a formula to calculate the sum of all proper divisors of a number?

I don't need to list all proper divisors, I just want to get its sum. Because for a small number, checking all proper divisors and adding them up is not a big deal. However, for a large number, this ...
17
votes
4answers
4k views

How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?

How can I prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$, without using prime decomposition? I should only use definition of gcd, division algorithm, Euclidean algorithm and corollaries to those. ...
17
votes
4answers
720 views

Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?

Let $p$ be a prime number. Show that $p^p-(p-1)^{p-1}$ can't be a square. In other words, there is no $n\in\mathbb{N}^{+}$ such that $$p^p-(p-1)^{p-1}=n^2.$$
17
votes
3answers
894 views

Prove that the number 14641 is the fourth power of an integer in any base greater than 6?

Prove that the number $14641$ is the fourth power of an integer in any base greater than $6$? I understand how to work it out, because I think you do $$14641\ (\text{base }a > 6) = ...
17
votes
2answers
797 views

Large integer help

The integer $5685858885855807765856785858569666876865656567858576786786785^{22}$ has 6436343 divisors. Using only a scientific calculator, find a way to show it has exactly 5 prime divisors.
17
votes
6answers
645 views

Calculate the 146th digit after the decimal point of $ \frac{1}{293} $

The question is: Calculate the 146th digit after the decimal point of $\frac{1}{293}$ 1 / 293 = 0,00341296928.., so e.g., the fifth digit is a 1. We know that 293 is a prime, probably this would ...
17
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4answers
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How do I compute $a^b\,\bmod c$ by hand?

How do I efficiently compute $a^b\,\bmod c$: When $b$ is huge, for instance $5^{844325}\,\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, ...
17
votes
2answers
923 views

Continued fraction for $\frac{1}{e-2}$

A couple of years ago I found the following continued fraction for $\frac1{e-2}$: $$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$ ...
17
votes
3answers
376 views

Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$ for positive integers $x$ and $y$?

Let $x$ and $y$ be positive integers. Is it possible that $(x+2)(x+1)x=3(y+2)(y+1)y$? I ran a program for $1\le{x,y}\le1\text{ }000\text{ }000$ and found no solution, so I believe there are none.
17
votes
2answers
3k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
17
votes
3answers
160 views

Is the numerator of $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\binom{n}{k}$ a power of $2$?

I stumbled on something numerically, and was just starting to work on it, but it seemed fun enough to share. Let $$f(n)=\sum_{k=0}^{n} \frac{(-1)^{k}}{2k+1}\binom{n}{k}$$ It appears, from the ...
17
votes
3answers
587 views

Improvement IMO 1988 $f(f(n))=n+1987$

The following problem was given at IMO 1987. Prove that there is no function f from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. So I tried to ...
17
votes
4answers
497 views

Do there exist infinitely many pairs of primes $(p,q)$ such that $pq$ divides $2^{p-1}+2^{q-1}-2$?

A mathematician friend gave me this question (partly as a joke) a few months ago and it has puzzled me for a long time:- Do there exist infinitely many pairs of primes $(p,q)$ such that ...
17
votes
1answer
647 views

Primes of the form $x^2 +ny^2$ where swapping $x$ and $y$ still gives a prime

I am studying primes of the form $x^2+ny^2$, and i was wondering if there are any known results about the primes of this form such that when you swap $x$ and $y$ you also get a prime. ie for ...
17
votes
2answers
714 views

Multiplication tables with all entries distinct

Let positive integers $\alpha$ and $\beta$ be given. It is easy to find sets $A$ and $B$ of positive integers such that: $|A|=\alpha$ and $|B|=\beta$ The set $P = \{ab : a\in A, b\in B\}$ contains ...
17
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2answers
2k views

what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by 47?

Can any one please tell the approach or solve the question what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by $47$? I can solve remainder of $45!$ divided by $47$ using Wilson's ...
17
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2answers
503 views

Is there an automated way to prove really boring elementary number theoretic results?

Motivation: I'm writing a proof, and within it, I need to prove: Conjecture: Let $p$ be an odd prime (i.e. $p \neq 2$). Let $c \geq 2$, $d \geq 1$ and $r \geq 1$. If $p^r$ divides $cd$, then ...
17
votes
2answers
526 views

Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (link to ...
17
votes
2answers
137 views

If two integer triples have the same sum of 6th powers, then their sums of squares agree $\bmod 9$

Given $$a^6 + b^6 + c^6 = x^6 + y^6 + z^6$$ prove that $$a^2 + b^2 + c^2 - x^2 - y^2 - z^2 \equiv 0 \bmod{9}$$ I was thinking of using $n^6 \pmod{27}$ and showing both sides have the same pattern ...
16
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13answers
4k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
16
votes
5answers
1k views

Prove that e is irrational

Prove that e is an irrational number. Recall that e $=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\mathrm{e}$ is rational. Then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = ...
16
votes
6answers
14k views

Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $

I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy. From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from ...
16
votes
5answers
2k views

Prove that every number ending in a $3$ has a multiple which consists only of ones.

Prove that every number ending in a $3$ has a multiple which consists only of ones. Eg. $3$ has $111$, $13$ has $111111$. Also, is their any direct way (without repetitive multiplication and ...
16
votes
2answers
1k views

To prove that $ [n/1]+ [n/2]+[n/3]+\dots +[n/n]+[\sqrt{n}]$ is even. [duplicate]

Let $n$ be a natural number. How do you prove that $$ \lfloor n/1 \rfloor+ \lfloor n/2\rfloor+ \lfloor n/3\rfloor+\dots +\lfloor n/n]+\lfloor \sqrt{n}\rfloor$$ is even? Thanks.
16
votes
6answers
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Prove $a+b+c+d $ is composite

Let $a,b,c,d$ be natural numbers with $ab=cd$. Prove that $a+b+c+d$ is composite. I have my own solution for this (As posted) and i want to see if there is any other good proofs.
16
votes
4answers
3k views

Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates?
16
votes
5answers
856 views

Puzzle: Cumulative Sum Divisible by 10

If we sum the first $4$ positive integers, we get $4 + 3 + 2 + 1 = 10$, which I think is pretty cool. I'm interested in seeing solutions to the following puzzle: If we take the cumulative sum of the ...
16
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5answers
2k views

Can n! be a perfect square when n is an integer greater than 1?

Can n! be a perfect square when n is an integer greater than 1?
16
votes
2answers
142 views

For $N\in \mathbb{N}$, do there exist natural numbers $N<n_1<n_2<\cdots<n_k$ such that $\frac{1}{n_1}+\cdots+\frac{1}{n_k}=1$?

$N$ is a natural number. Is there any $k$ and some natural numbers $N<n_1<n_2<\cdots<n_k$ such that $$\frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_k}=1$$?
16
votes
5answers
957 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
16
votes
1answer
398 views

All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 ...
16
votes
3answers
555 views

Solving $p_1^{e_1} p_2^{e_2}…p_k^{e_k}=e_1^{p_1} e_2^{p_2}…e_k^{p_k}$

Find all positive integers $k$, positive integers $e_i$, and distinct prime numbers $p_i$ for $1\le i\le k$, such that $$p_1^{e_1} p_2^{e_2}...p_k^{e_k}=e_1^{p_1} e_2^{p_2}...e_k^{p_k}.$$ Is this ...
16
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5answers
1k views

Bézout's identity proof that if $(a,b,c)=1$ then $ax+bxy+cz=1$ has integer solutions

Massive edit to simplify the question. Some comments below might be made obsolete - specifically, the comment that this follows directly from Dirichlet. That was true for the original wording. I'm ...
16
votes
2answers
432 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
16
votes
4answers
317 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 ...
16
votes
2answers
460 views

Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
16
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1answer
2k views

Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz

The question is motivated by a question on a can of number noodles. Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
16
votes
2answers
339 views

On $a^4 + b^4 = c^4 + d^4 = e^5$.

Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.
16
votes
1answer
202 views

How to sum this infinite series

How to sum this series: $$\frac{1}{1}+\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$$ My attempt: Multiply and divide the series by $9$ ...
16
votes
2answers
931 views

$(x-a)(x-b)(x-c)(x-d)=ex$

We can verify that $x=125,162,343$ are the roots of equation $(x-105)(x-210)(x-315)=2584x$. My question is,Could you find five positive integers $a,b,c,d,e$, which $(x-a)(x-b)(x-c)(x-d)=ex$ has four ...
15
votes
18answers
4k views

How to prove $n^5 - n$ is divisible by 30 without reduction

How can I prove that prove $n^5 - n$ is divisible by 30? I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$ Now, $n(n-1)(n+1)$ is divisible by 6. Next I need to show that $n(n-1)(n+1)(n^2+1)$ is ...
15
votes
4answers
2k views

Proof that one large number is larger than another large number

Let $a = (10^n - 1)^{(10^n)}$ and $b=(10^n)^{(10^n - 1)}$ Which of these numbers is greater as n gets large? I believe it is $a$ after looking at some smaller special cases, but I'm not sure how to ...
15
votes
5answers
3k views

Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?

About this Fibonacci sequence, is it true that the remainders when divided by three repeat along with the sequence like this: Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
15
votes
14answers
2k views

Name of the highest power of 2 smaller than or equal to a given number

For a number $x$, I would like to know whether there is a common name for the number $2^n$ such as $2^n \leq x < 2^{n+1}$ (e.g. If $x = 7$, then $2^n = 4$, $n = 2$). I have some computer science ...
15
votes
7answers
726 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
15
votes
10answers
9k views

What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi (A\cdot B)=\varphi (A)\cdot \varphi (B)$, if A and B are coprime? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its ...