6
votes
2answers
89 views

What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)

(The question arises from playing with translating series into integrals) I wanted to see, what it means to have a "continuous" relative for powerseries and other series; the most simple one ...
3
votes
0answers
46 views

Josephus Variant

I set myself the challenge of trying to solve a variant of trying to solve a variant of the josephus problem where instead of killing every second person, every third person dies. The formula for the ...
0
votes
0answers
24 views

What are some alternatives to base number systems and their advantages?

So apparently the introduction of base number systems was great. But are there other systems which might have uses for other things? A an example consider a system where each digit has value n! and ...
0
votes
0answers
65 views

Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
5
votes
1answer
79 views

Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
1
vote
0answers
55 views

A Problem for the year with prime decomposition

I have noticed (and hope there are no errors) that: $$2013=3\times 11\times 61$$ $$2014=2\times 19\times 53$$ $$2015=5\times 13\times 31$$$ while 2012 and 2016 are not the product of exactly 3 ...
8
votes
1answer
85 views

Range for values of cyclotomic polynomials, where $x$ is replaced by the golden ratio $0.61…$ ? And is it dense?

This is a recreational math question. I just played with the cyclotomic polynomials; and replacing $x$ by $1$,$-1$,$I$ gives some interesting patterns; setting $x=2$ seems to give some ...
0
votes
2answers
190 views

A problem for math lovers to count the digits

Today a classmate of mine asked a question which is based on counting. Question. Find a positive integer which when multiplied up to $N$ times will give numbers having the same digits but differently ...
4
votes
2answers
122 views

$2^5 \cdot a^b=2,5ab$

I came across this problem in an elementary number theory book, and I think I solved it. Well, the question is posed as $2^5 \cdot 9^2 = 2,592$. Are there any other pairs $a,b \in \mathbb{Z}$ such ...
7
votes
1answer
58 views

Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
9
votes
1answer
169 views

A congruence in the number of certain ternary strings

Let $a_n$ be the number of ternary strings of length $n$ which do not contain three consecutive symbols that are all different. That is, $$a_n = \Bigl|\bigl\{\,(b_k)_{1\leq k\leq n}\in ...
31
votes
1answer
329 views

Zero-avoiding integers

Let's say an integer $n>2$ is zero-avoiding if, for every $2\leq b < n$, the representation of $n$ in base $b$ has no $0$ digits. (Obviously every $n$ has a $0$ when written in base $n$ and no ...
3
votes
3answers
79 views

Is there a $3\times 3$ magic square adding up to $7$.

I suspect that there is no magic square with natural number entries (matrix where each row, column and long diagonal add up to the same number) which would add up to $7$. There is no restriction on ...
6
votes
3answers
183 views

If $x^3+\frac1{x^2}=1$, what is $x^3+\frac1{x^3}$?

$x^3 + \frac1{x^2} = 1$. Then, $x^3 + \frac1{x^3} = ~?$ $p + \frac1{p^2} = 47$. Then, $p + \frac1p = ~?$
1
vote
2answers
169 views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
8
votes
4answers
327 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
20
votes
2answers
345 views

The next number that has this property?

I noticed that $1/8 = 0.125$ and the sum of the digits of the number $0.125$ is $0+1+2+5=8$. It's lovely. I searched other numbers who have that propriety : I only found $1$, $3$ and $8$. Is there ...
1
vote
0answers
60 views

Existence of a Vampire number on the form $v = xy = a^bb^a$?

A number $v = xy$ with an even number $n$ of digits formed by multiplying a pair of $n/2$-digit numbers (where the digits are taken from the original number in any order) $x$ and $y$ together. ...
1
vote
2answers
96 views

Is $\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}?$, for $a,b,c,d>0$?

Is $$\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}$$ for $a,b,c,d>0$ If it is true, then can we generalize? EDIT:typing mistake corrected. EDIT, WILL JAGY. Apparently the real question is Is ...
31
votes
2answers
751 views

Proof that $123456789098765432111$ is prime?

The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact $$12345678987654321 = ...
8
votes
3answers
711 views

What is the probability that GCD of $(a,b)$ is $b$?

My question is quite simple. I have been googling a lot lately trying to find a solution to this: Given a sequence of n integers $[1,2,...,n]$. If we pick two numbers randomly from the set say, a and ...
-6
votes
5answers
132 views

Construct natural numbers $1$-$100$ using $\pi$ [closed]

Here are the rules of the game: Using only the following operators and numbers construct the natural numbers from 1-100 Operators: only one $\lfloor\space\rfloor$ xor $\lceil\space\rceil,(\space), ...
6
votes
2answers
202 views

$20$ hats problem [duplicate]

I've seen this tricky problem, where $20$ prisoners are told that the next day they will be lined up, and a red or black hat will be place on each persons head. The prisoners will have to guess the ...
1
vote
0answers
85 views

gcd finding method

An integer $d$ is a $\gcd$ of two non-zero integers $a$ and $b$, if $d$ divides $a$ & $d$ divides $b$ '$c$ divides $a$ & $c$ divides $b$' implies '$c$ divides $d$' for any integer $c$. If ...
1
vote
0answers
33 views

Are there infinitely many emirps? [duplicate]

An emirp is a prime number such that when its decimal digits are reversed, one obtains a different prime number. Are there infinitely many ermips? It is apparently open whether there are infinitely ...
3
votes
2answers
95 views

Does this process always terminate?

Consider the following "game". Take two natural numbers $n \leq m$ and let $S=n+m$ and $P=nm$. Take two logicians A and B, and tell A the value of $S$ and B the value of $P$. Now, A and B alternate ...
23
votes
1answer
305 views

The final number after $999$ operations.

I wanted to know, let the numbers $1,\frac12,\frac13,\dots,\frac1{1000}$ be written on a blackboard. One may delete two arbitrary numbers $a$ and $b$ and write $a+b+ab$ instead. After $999$ such ...
7
votes
2answers
2k views

Is there any mathematical theory behind sudoku?

In particular I would like to know: is it possible to say if a sudoku is solvable only having the initial scheme? If yes, what are the condition for which it is solvable? Given the initial scheme of ...
2
votes
1answer
121 views

kaleidoscopic effect on a triangle

Let $\triangle ABC$ and straightlines $r$, $s$, and $t$. Considering the set of all mirror images of that triangle across $r$, $s$, and $t$ and its successive images of images across the same ...
4
votes
2answers
135 views

Four integer numbers to express all integers from 1 to 40 [duplicate]

Let $a$, $b$, $c$, and $d$ four integers such that $0 <a <b <c <d$. We can get all integers from $1$ to $40$ by expressions containing or not only the numbers $a, b, c$ and $d$. In these ...
33
votes
4answers
1k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
4
votes
4answers
196 views

Find the largest number having this property.

The $13$-digit number $1200549600848$ has the property that for any $1 \le n \le 13$, the number formed by the first $n$ digits of $1200549600848$ is divisible by $n$ (e.g. 1|2, 2|12, 3|120, 4|1200, ...
5
votes
2answers
238 views

Let $k \geq 3$; prove $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$

Prove: If $k \geq 3$, then $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$, where $k, m, n \in \mathbb{N}$.
6
votes
4answers
613 views

Proving that none of these elements 11, 111, 1111, 11111…can be a perfect square [duplicate]

How can i prove that no number in set S S = {11, 111, 1111, 11111...} Is a perfect square. I have absolutely no idea how to tackle this problem i tried rewriting it in powers of 10 but that didn't ...
3
votes
1answer
99 views

Minimum number of coconuts

Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the ...
2
votes
3answers
156 views

$2^n-3^m=1 , m,n \in \mathbb N =?$

$2^n-3^m=1 , m,n \in \mathbb N =?$ my questions are: do m,n exist? are they finitely many $m,n$? if there are infinitely many is there a way to describe them all? Same question about $3^n-2^m=1 $, ...
2
votes
1answer
106 views

A game involving points in the integer plane - who wins?

I am running a workshop on puzzles and problem solving over the weekend and thought that it might be a good idea to get people engaged by phrasing some interesting mathematical results in terms of ...
5
votes
1answer
184 views

Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed?

Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed? We define $f(n)=m$ where the digits of $m$ and $n$ are reverse. Such as ...
31
votes
1answer
626 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?
5
votes
3answers
285 views

Prove that $n+1$ elements of a set will contain a co-prime pair

Suppose $P$ is a set of $n + 1$ integers selected from $1,2,3,...,2n + 1$. Then how can we show $P$ contains two coprime integers? The result holds if $P$ contains only $n$ integers?? Added Let ...
6
votes
5answers
197 views

How can I find the value of $a^n+b^n$, given the value of $a+b$, $ab$, and $n$?

I have been given the value of $a+b$ , $ab$ and $n$. I've to calculate the value of $a^n+b^n$. How can I do it? I would like to find out a general solution. Because the value of $n$ , $a+b$ and $ab$ ...
-2
votes
1answer
92 views

Solve for $x$: $\sqrt{12} - \sqrt[3\leftroot1]{720} = \sqrt{x}$

I want to solve for $x$ Here's the question $$\large \sqrt{12} - \sqrt[3\leftroot1]{720} = \sqrt{x}$$ I need to find the value of $x$ Help!
5
votes
3answers
157 views

Does $p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n)$ imply $f(m+n)=f(m)+f(n)$?

Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that: $$(\forall p: \mathrm{~prime~})(\forall m,n\in\mathbb{N})(p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n))$$ is $f$ linear? by linear I mean: ...
3
votes
2answers
273 views

Conjecture I came up with

For each number translated into binary $0$, $1$, $10$, $11$, $100$, $101$, $110$, $111$, $1000$, ... find a number where, when you take the length of the binary number, the binary number and the ...
74
votes
1answer
3k views

$4494410$ and friends

The number $4494410$ 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. ...
2
votes
1answer
254 views

finding nearest perfect square for a fractional number

Is it possible to have a clear definition for the nearest perfect square number for a fractional number? For example, let us consider a number 0.004. What is another decimal number closest to it, that ...
6
votes
1answer
235 views

How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?

So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this: ...
8
votes
1answer
189 views

Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Consider a modified version of Collatz sequence: $C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$ Let $F_n$ be the ...
1
vote
1answer
343 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
2
votes
0answers
111 views

For which positive integers n does there exist a prime whose digits sum to n?

Motivated by this earlier question, I thought of this problem: Question: For which positive integers $n$ does there exist a prime whose decimal digits sum to $n$? We can make two "easy" ...