Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.
152
votes
13answers
17k views
Unusual 5th grade problem, how to solve it
Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was ...
143
votes
1answer
6k views
How many fours are needed to represent numbers up to $N$?
The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols.
For example, $165=(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}) ...
122
votes
2answers
12k views
Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual
An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, ...
120
votes
4answers
4k views
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It ...
92
votes
6answers
17k views
Deleting any digit yields a prime… is there a name for this?
My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100:
But ...
73
votes
8answers
2k views
What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
57
votes
4answers
5k views
What is special about the numbers 9801, 998001, 99980001 ..?
Just saw this post, and realized that
1/9801 =
...
43
votes
7answers
5k views
What makes $9$ special?
I don't know if this is a well know fact but I have observed that every number no matter how large that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there ...
42
votes
2answers
1k views
Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on
Popular mathematics folklore provides some simple tools
enabling us compactly to describe some truly enormous
numbers. For example, the number $10^{100}$ is commonly
known as a googol,
and a googol
...
41
votes
6answers
727 views
Second part of the factorial sum divisibility question
Which primes $p$ divide the sum of factorials $1! + 2! + 3! + 4! + 5! + \cdots + (p-1)!$? This is related to my previous question.
40
votes
4answers
2k views
$x$, $y$, $x+y$ and $x-y$ are prime numbers. What is their sum?
Here is the question:
The $x$, $y$, $x−y$ and $x+y$ are all positive prime integers. What is the sum of all the four integers?
Well, I just put some values and I got the answer.
$x=5$, $y=2$, ...
39
votes
4answers
1k views
How to understand and appreciate the prime number industry?
Why would I want to buy prime numbers? There is a website (found it!) selling a table of 400 digit primes for twenty dollars. Like an updated version of this. I have a layman's idea that prime numbers ...
38
votes
4answers
2k views
Prove every odd integer is the difference of two squares
I know that I should use the definition of an odd integer ($2k+1$), but that's about it.
Thanks in advance!
37
votes
3answers
1k views
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]
Possible Duplicate:
Highest power of a prime $p$ dividing $N!$
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
36
votes
7answers
783 views
Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.
Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows:
$x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
34
votes
8answers
2k views
32
votes
6answers
2k views
Is $2^{218!} +1$ prime?
Prove that $2^{218!} +1$ is not prime.
I can prove that the last digit of this number is $7$, and that's all.
Thank you.
32
votes
1answer
1k views
Decomposing polynomials with integer coefficients
Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
31
votes
7answers
2k views
Project Euler, Problem #25
Problem #25 from Project Euler asks:
What is the first term in the Fibonacci sequence to contain 1000 digits?
The brute force way of solving this is by simply telling the computer to generate ...
30
votes
7answers
4k views
Why is every answer of $5^k - 2^k$ divisible by 3?
We have the formula $$5^k - 2^k$$
I have noticed that every answer you get from this formula is divisible by 3. At least, I think so. Why is this? Does it have to do with $5-2=3$?
30
votes
9answers
2k views
Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.
I'm 16 years old, and I'm studying for my exam maths coming this monday. In the chapter "sequences and series", there is this exercise:
Prove that a positive integer formed by $k$ times digit 1, ...
30
votes
1answer
550 views
Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers
For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
29
votes
2answers
700 views
Any positive integer solutions to $x^6+y^{10}=z^{15}$?
This question might be easy.
The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, ...
28
votes
7answers
900 views
Bad Fraction Reduction That Actually Works
$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$
This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
27
votes
5answers
3k views
Are all prime numbers finite?
If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid ...
27
votes
6answers
580 views
Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
25
votes
4answers
558 views
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime [duplicate]
The number $30$ has a curious property:
All numbers co-prime to it, which are between $1$ and $30$ (non-inclusive) are all prime numbers!
I tried searching(limited search, of course) for numbers ...
25
votes
1answer
586 views
Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $(m-n) | (f(m)-f(n))$
A long time back, I wondered what functions other than integer polynomials on $\mathbb{N}$ (or $\mathbb{Z}$) satisfied the property:
$$\forall m,n: (m-n) | (f(m)-f(n))$$
Turns out that, on ...
23
votes
2answers
1k views
Proof of recursive formula for “fusible numbers”
The set of fusible numbers is a fantastic set of rational numbers defined by a simple rule. The story is well told here but I'll repeat the definitions. It's the formula on slide 17 that I'm trying to ...
23
votes
0answers
845 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
22
votes
4answers
1k views
Is the statement “1/3 of the natural numbers are divisible by 3” true? Is anything similar to it true?
If we're talking about a finite set of the natural numbers, like those between 1 and 500 or 1 and a million, it seems to me that the fraction of numbers in that finite set that have a factor of 5 ...
22
votes
4answers
468 views
Elementary proof of $m^n\neq n^m$ for almost all natural numbers $m\neq n$
$2^4=16=4^2$. In fact, $\{2,4\}$ is the only pair of natural numbers with that property, i.e. if $m<n$ are natural numbers and $m^n=n^m$, then $m=2$ and $n=4$.
This is easily seen with some ...
22
votes
3answers
490 views
Intuition on fundamental theorem of arithmetic
I'm sorry ahead if time if this is overly trivial for this site.
Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my ...
22
votes
2answers
443 views
When is a sum of consecutive squares equal to a square?
We have the sum of squares of $n$ consecutive positive integers: $$S=(a+1)^2+(a+2)^2+ ... +(a+n)^2$$ Problem was to find the smallest $n$ such, that $S=b^2$ will be square of some positive integer. I ...
22
votes
1answer
591 views
Always oddly-many ones in the binary expression for $10^{10^{n}}$?
Update: Pending independent verification, the answer to the title question is "no", according to a computation of $q(10) = 11609679812$ (which is even).
Let $q(n)$ be the number of ones in the ...
22
votes
2answers
454 views
How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number?
How to prove that $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$?
Thanks in advance.
21
votes
3answers
2k views
Yitang Zhang: Prime Gaps
Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific.
EDIT$^1$:
Are there any experts here who can ...
20
votes
9answers
3k views
Prove $2^{1/3}$ is irrational.
Please correct any mistakes in this proof and, if you're feeling inclined, please provide a better one where "better" is defined by whatever criteria you prefer.
Assume $2^{1/2}$ is irrational.
...
20
votes
8answers
686 views
If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
when, $a=3$
we have, $a+2=5$ and $a+4=7$
20
votes
4answers
1k views
Why is the last digit of $n^5$ equal to the last digit of $n$?
I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ...
20
votes
3answers
334 views
Why does $10^x$ have $(x+1)^2$ factors?
eg. $1000$ has $16$ factors $(1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000)$
20
votes
1answer
262 views
For what values of $n$ is $n^2+n+2$ a power of $2$?
Working on isometric paths in hypercubes, I came up with the following simple, yet (imo) interesting problem. For what natural numbers $n$ exists a natural number $t$ such that $n^2+n+2=2^t$? The ...
20
votes
1answer
479 views
A fun Pascal-like triangle
A while back, inspired by Pascal, I put on some shackles and a thorny belt. Soon after, inspiration came pouring in, and I thought of the following triangle:
$$
\begin{array}{rcccccccccc}
& ...
20
votes
4answers
529 views
To what extent can values of $n$ such that $n^2-n+41$ is composite be predicted?
Euler's polynomial $E(n)=n^2-n+41$ takes a prime value for each of the positive integers $n<41$. For $n=41$, its value is $41^2$, which is composite, and every multiple of 41 will likewise produce ...
19
votes
1answer
468 views
prime as sum of three numbers whose product is a cube
Good evening!
I am very new to this site. I would like to put the following materiel from Prof. Gandhi's note book and my observations. Of course it is little long with more questions. But, with good ...
18
votes
8answers
727 views
Comparing $2013!$ and $1007^{2013}$
I have to compare the following two numbers:
$$2013! \text{ and } 1007^{2013}$$
where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$.
I tried in different ways to group the $1 \times 2 ...
18
votes
1answer
286 views
Infinitely many primes of the form $\lfloor \sqrt {3} \cdot n \rfloor $?
How to prove or disprove following statement :
There are infinitely many primes of the form : $\lfloor \sqrt {3} \cdot n \rfloor $
Note: This is a problem I made myself.
There is a theorem ...
18
votes
1answer
620 views
Did Leonardo of Pisa prove $n=4$ case of FLT?
Reputable on-line sources agree that Leonard 'Fibonacci' proved the nonexistence of positive-integer solutions to $c^4 - b^4 = a^2$ . Yet my change to Wikipedia to reflect this was reverted. I hope ...
17
votes
9answers
3k views
Do infinity and zero really exist?
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please ...
17
votes
2answers
687 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.
