Tagged Questions

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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 ...
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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 ...
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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 ...
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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 ...
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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'' ...
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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 ...
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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 $2S$ ...
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$12345679$ 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 ...
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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 ...
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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.
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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. $c(2,2) = 22$ $c(8,9) = 89$ ...
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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 ...
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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.
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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...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $O(\log N)$.
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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 ...
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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, ...
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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: ...
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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 ...
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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 ...
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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???
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
$2013$ 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 ...