1
vote
0answers
48 views

How do you evaluate $a^b$ where b is irrational using only basic operators.

How would you evaluate $a^b$ where b is irrational and you can only use +,-, multiplication, division, and rational powers. For example $2^\pi.$ We know $2^2$ = $2\times2$ etc... but when the power ...
1
vote
1answer
73 views

Prove that in any base the number of digits composing the repetitive mantissa of the reciprocal of a prime $p$ never exceeds $p-1$.

I was trying to find bases where the reciprocals of primes have a short repetitive mantissa. Here is what I found: http://imagizer.imageshack.us/a/img835/7738/c7gb.png The bases are on the left. The ...
3
votes
1answer
61 views

Interesting sequence question

I saw a puzzle the other day and it was as follows: Find the next number in the sequence: 1 11 21 1211 111221 312211 ... If anyone wants to have a go at the ...
0
votes
1answer
45 views

How often does $p^k$ divide the Fibonacci numbers?

I would like to know about the Fibonacci numbers $F_n = 1,1,2,3,5,8, \dots$ in $\mathbb{Z}/p^k\mathbb{Z}$. $$ \mathbb{P}[p^k \text{ divides } F_n ] = \frac{\#\{1 \leq n\leq N: F_n \equiv 0 \mod ...
2
votes
1answer
70 views

Getting multiple of 11

Take any $2$ digit number having different digits. Now add the bigger digit in that number. By continuing this process, you will get a multiple of $11$ i.e. both of the digits will be equal of a ...
0
votes
2answers
57 views

A question of divisibility.

Let $p$ and $q$ are relatively prime integers. Consider $S = \{\frac{p}{q}\} + \{\frac{2p}{q}\} + \{\frac{(q-1)p}{q}\}$, where $\{x\}$ is the fractional part of the real number $x$. Prove that $2S$ ...
10
votes
2answers
557 views

$12345679$ and friends

We can see that in the decimal system each of $12345679\times k$ $(k\in\mathbb N, k\lt 81, k\ \text{is coprime to $9$})$ (note! not $123456789$) has every number from $0$ to $9$ except one number as ...
2
votes
0answers
390 views

Sum of cubes of the digits of a number equal to to the number

I have a number, I don't know how large or small, but if I cube the digits of the number and sum them, the sum is equal to the number itself. In other words, $$\sum_{k=1}^n{a_k^3}=\sum_{k=1}^{n}{a_k ...
6
votes
0answers
221 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
3
votes
0answers
176 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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 ...
1
vote
1answer
50 views

Any other prime numbers that satisfy this condition?

If $a=2$ and $b=3$, then $a^2-1$ is an integer multiple of $b$. Is there any other pair of primes $a$ and $b$ that satisfy this relationship? I don't think so, but can't figure out why not.
1
vote
2answers
78 views

Is there an algebraic method to concat two numbers?

I'm searching an algebraic way to concat numbers in base $10$. Concatening two numbers is to put side by side their notations. Let $c$ a concatenating function. $c(2,2) = 22$ $c(8,9) = 89$ ...
7
votes
1answer
200 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
1
vote
3answers
644 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.
3
votes
1answer
74 views

Minimal diameter of set of fractions

Let $p_n$ be a pairwise partition of $\{1,2,...,2n\}, n\in \bf N$ where $(a,b)\in p \implies a<b$, and $P_n$ the set of all such pairwise partition. $d(n) := \min_{p_n\in ...
5
votes
1answer
83 views

Investigating the density of special integers

I was working on a problem earlier and came up with a solution using the following type of integers: Call an integer $n \geq 3$ convenient if the following hold: $n$ is a squarefree odd integer if ...
3
votes
1answer
235 views

My conjecture on almost integers.

Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x ...
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?
33
votes
2answers
820 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
741 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 ...
4
votes
1answer
113 views

Fictional math proof = prime return function

I am trying to write a piece of future fiction where one of the characters is famous for proving an important truth related to primes. I want to make it as realistic as possible, but i'm not a math ...
3
votes
1answer
75 views

Basic number theory curiosity

Let $a,b $ be positive integers, and consider the set $D := \{am + bn: m,n \in \mathbb{Z} $ and $am + bn > 0 \}$. Let $d$ be the minimum integer in $D$. Is it true that $D = \{kd : k \in ...
2
votes
3answers
152 views

How many zeros are there in $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$?

How many zeros are there at the end of $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$ ? I tried it by grouping all the $2$'s and $5$'s and $5$'s and $6$'s but cant get my answer...
1
vote
0answers
87 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 ...
6
votes
0answers
263 views

Can $ 262537412640768743.99999999999925 $ be beaten with simple expressions? [closed]

We know: $$\begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align}$$ where $x$ is the real solution ...
3
votes
1answer
104 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 ...
5
votes
3answers
319 views

Distribution of palindromic numbers

We all know what a palindromic number is, it is a number which is the same, independent from which side we read it, for example 101, 202, 33733,.... It is also clear that there are infinity many ...
3
votes
2answers
130 views

Recreational number theory problem

Suppose we have a positive integer $n$ that has exactly three distinct prime factors, say $p,q, r$. How can we find a formula for the number of positive integers $\leq n$ that are divisible by none of ...
3
votes
2answers
86 views

Number theory Exercise

for positive integer $n$, how can we show $$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$ where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square ...
35
votes
2answers
849 views

Do all natural numbers have a nonzero multiple that is a palindrome in base 10?

Some natural numbers have a nonzero multiple that is a palindrome in base 10. For example, $106 \times 2 = 212$, which is a palindrome, and $29 \times 8 = 232$, which is also a palindrome. Aside ...
2
votes
1answer
613 views

How to calculate $ 1^k+2^k+3^k+\cdots+N^k $ with given values of $N$ and $k$? [duplicate]

Here $ 1<N<10^9$ and $0<k<50$ So we have to calculate it in order of $O(\log N)$.
1
vote
1answer
123 views

Primes in a certain arithmetic progression

Primes $=_m 1 $. For any positive integer m, prove that arithmetic progression $$1 +m, 1 + 2m, 1 + 3m, ... $$ contains infinitely many primes. How can we prove that it suffices to show that for ...
3
votes
1answer
263 views

Triangle from a given rectangle

We are given a set of (marked) points in a 2D coordinate system and function $f(x,y)$ which counts number of points marked in the rectangle $(0 , 0), (x , y)$ - where $(0 , 0)$ if down-left corner, ...
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
336 views

Are numbers with repeating patterns in their decimal expansion (e.g. $0.123123123\ldots$) rational?

There's a question that I've been thinking about for quite some time now. We all know that numbers with infinite decimal expansion such as $0.\overline{3}$ or $0.\overline{1}$ are not necessarily ...
3
votes
4answers
193 views

Are $4ab\pm 1 $ and $(4a^2\pm 1)^2$ coprime?

Let $a\ne b$ be two positive integers. Are $4ab+1$ and $(4a^2+1)^2$ coprime always? Can you find $a$ and $b$ such that they are not coprime? Edit: It has been proved that $4ab-1$ is not a divisor ...
10
votes
4answers
295 views

Number theory fun problem

Say $a,b > 2 $ are integers. Then we have that $2^a + 1$ is not divisible by $2^b - 1$. Any thoughts on how to tackle this problem???
0
votes
3answers
121 views

Given $n+1\mid2\sum_{k=1}^{n}{a_k}$, find $a_k$.

Let $m$ be a positive integer. There are only 2 finite sequences of positive integers like $a_1,a_2,...,a_m$ such that $$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
3
votes
2answers
275 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 ...
5
votes
2answers
625 views

Sailors, monkey and coconuts

Five sailors and a monkey were shipwrecked on a deserted island, and they spent the first day gathering coconuts for food, piled them all up together and went to bed. But when they were all asleep one ...
9
votes
1answer
206 views

Is the number of alternating primes infinite?

I'm not sure if the recreational-mathematics tag is appropriate, but this problem came up during a practice Putnam seminar so maybe? The problem: Say that a positive integer is alternating if ...
1
vote
2answers
295 views

How many ways to write one million as a product of three integers?

In how many ways can the number 1;000;000 (one million) be written as the product of three positive integers $a, b, c,$ where $a \le b \le c$? (A) 139 (B) 196 (C) 219 (D) 784 (E) None of the ...
11
votes
1answer
260 views

The number of prime years in a lifetime

$2013$ is not a prime: $3 \times 11 \times 61$. I was born in a prime year, and if I live as expected according to the statistics for U.S. males, I will just reach another prime year, $2027$. That ...
6
votes
2answers
750 views

Useless math that become useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless. My idea is to amend my article with some theories that seemed useless when they are created but ...
5
votes
1answer
217 views

asymptotics of the square indicator sum for subsequences of digits

Introduce the function $g(n)$ defined for positive integer arguments $n$ as the count of squares of positive integers among the numbers that can be formed by taking some subsequence of the digits of ...
0
votes
1answer
112 views

Gradually rising or falling numbers

I'm looking for a number series I can use for gradually rising or falling numbers. The number series should not be linear and should converge to a number at some point. (Sorry I'm really scared of ...
0
votes
0answers
135 views

Is there an emirp greater than $10^{10006}+941992101 \times 10^{4999}+1$?

An emirp is a prime such that a distinct prime is formed when its digits are reversed. According to Wikipedia (and its references), the largest known emirp is \[p:=10^{10006}+941992101 \times ...
10
votes
3answers
1k views

Where does Feigenbaum's Constant (4.6692…) originate?

Feigenbaum discovered a ratio between bifurcations that were found in all known chaotic-dynamic systems, from dripping water faucets to abstract equations on population fluctuations (as elucidated in ...
5
votes
2answers
235 views

Sum(Partition(Binary String)) = $2^k$

So given any binary string B: $$b_1 b_2 \dots b_n$$ $$b_i \in \{0,1\}$$ It would seem it is always possible to make a partitioning of B: $$ b_1 b_2 \dots b_{p_1}|b_{p_1 + 1}b_{p_1 + 2}\dots ...