1
vote
0answers
34 views

Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit?

[This has been cross-posted to MO.] A positive integer $N$ is said to be a perfect number if $$\sigma(N) = 2N,$$ where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is perfect ...
3
votes
1answer
68 views

How can “Lucky Numbers” be approached rigorously?

To begin with, "Lucky Numbers" are a sequence of numbers generated by a sieve similar to the Sieve of Eratosthenes for finding primes. It starts with the set of natural numbers. Begin by selecting ...
0
votes
0answers
54 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
1
vote
1answer
71 views

Magic square of numbers

What is the logic behind filling up a magic square? I have understood the algorithm of filling up a magic square of 3*3 or 5*5. I really do not know how to derive this
3
votes
0answers
95 views

Interesting Number Game

Let's say you are given some random positive integer. Next, assume you are given the following sequence of integers: $$[1, 2, 3, 4, 5, 6, 7, 8, 9].$$ Now, assume you are allowed to use the following ...
2
votes
6answers
102 views

Investigating the linearity between squares and their roots

I recently noticed that $\sqrt{128} = 11.31$ and that $128$ is $\approx 30\%$ between $121 = 11^2$ and $144=12^2$, that is: $$ \frac{128-121}{144-121} = \frac{7}{23} \approx 30\%$$ and $\sqrt{128} = ...
7
votes
2answers
137 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
67 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
32 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
69 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
83 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
58 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
93 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
200 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
125 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
60 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
181 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
340 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
87 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
187 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
4answers
949 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.
9
votes
4answers
446 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 ...
21
votes
2answers
363 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
66 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
97 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 ...
33
votes
2answers
845 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
787 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 ...
18
votes
1answer
358 views

Does there exist a general solution of this 'Counting numbers' game?

A few days ago, a friend of mine taught me a number-game. It may be famous, but I haven't known it. I'm going to show it to you. Imagine that you have a kind of page-a-day calendar, and that you play ...
6
votes
2answers
233 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
88 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
103 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
319 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
127 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
142 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
205 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
240 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
892 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
110 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
113 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
215 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 ...
32
votes
1answer
639 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
318 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
200 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
95 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
159 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
276 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 ...