3
votes
1answer
88 views

Is there a prime number ending with the natural number $n$

if $n$ not is divisible by 2 or 5? Example: given 813075843967837637675737563754361301, there is a prime 20813075843967837637675737563754361301 or given ...
1
vote
0answers
46 views

Primes made from sequential digits

While messing around, I noticed that across some prime numbers contain only sequentially increasing digits, e.g. $23, 67, 89,23456789$. If we adopt a convention of returning to $1$ after a $9$, we ...
4
votes
0answers
78 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
22
votes
1answer
233 views

Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?

This past Wednesday's What-If had this image at the bottom: In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that ...
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'' ...
0
votes
2answers
77 views

how to find taxicab numbers but for squares?

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205. $$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 ...
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
75 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
65 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
46 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
82 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
58 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
588 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
627 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 ...
7
votes
0answers
226 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
179 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
116 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
52 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
84 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. $c(2,2) = 22$ $c(8,9) = 89$ ...
8
votes
1answer
213 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
4answers
950 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
76 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
90 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
239 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
477 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
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 ...
4
votes
1answer
115 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
156 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
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 ...
6
votes
0answers
267 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
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 ...
5
votes
3answers
337 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
132 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
887 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
685 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
126 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
281 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
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
357 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
318 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
124 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
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 ...
5
votes
2answers
705 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
216 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
317 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 ...
12
votes
1answer
264 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 ...