2
votes
1answer
27 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
18
votes
4answers
287 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
3
votes
2answers
202 views

Calculating $a^n\pmod m$ in the general case

It is well known, that $$a^{\phi(m)}\equiv1\pmod m ,$$ if $\gcd(a,m)=1.$ So, $a^n\pmod m$ can be calculated by reducing n modulo $\phi(m)$. But, for the tetration modulo $m$ $$a \uparrow ...
8
votes
0answers
79 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
2
votes
0answers
79 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
1
vote
1answer
52 views

Two variable integer equation

I have the following equation: $$ p^q(2^{q-1}-1)=9p^7q $$ I need to solve for $p$ and $q$. $p$ and $q$ are integers. I think I could take the case $p=0$ separately and for that one $q$ could be ...
3
votes
3answers
123 views

Proofs for $0^0 =1$? [duplicate]

Everyone knows the following: $$0^x = 0 \quad \wedge \quad x^0 = 1 , \quad\forall x \in R^*$$ One morning, I wake up asking myself the question "$\text{What is $0^0$, then?}$". So, I did what any ...
2
votes
3answers
61 views

computing $2^{170}+ 3^{63}\pmod {19}, 3^{175} + 2^{73} \pmod {17}$, etc… by hand

I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for ...
0
votes
1answer
55 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
8
votes
1answer
109 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
2
votes
1answer
64 views

Why inverse modulo exponentiation is harder than inverse exponentiation without modulo

I am new to number theory. I read in cryptography inverse modulo exponentiation is used because it is hard. But I couldn't understand the advantage of it over inverse exponentiation without modulo. ...
0
votes
2answers
70 views

Square roots and powers

This is a rather silly question. In what order does one evaluate a combination of powers and fractional powers? I have the question phrased: $\sqrt{ 1/4^2 }$ OR ${ 1/4 }$? Which is greater? I ...
3
votes
2answers
152 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
2
votes
1answer
54 views

Formula/Algorithn for Exponential factoring?

Given $s = a^b$ find $a$ and $b$. my first algorithm was the obvious brute force method of checking all $b$ roots or dividing by all possible $a$. But I am wondering if there is a more efficient ...
0
votes
5answers
304 views

Generate solutions of Quadratic Diophantine Equation

Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation Here's the answer: $$ ...
2
votes
1answer
74 views

Imperfect digit-to-digit invariants in Base $10$

$3435 = 3^3 + 4^4 + 3^3 + 5^5$ is an example of a perfect digit-to-digit invariant. Fact: The number of PDDIs is finite for any given base; in particular, for base $10$. Question: Working over base ...
3
votes
2answers
87 views

Is there an easy way to find $2^p \pmod p$?

I'm working on a number theory problem, and was able to simplify an expression down to a quadradic in $2^p \pmod p$? Is there an easy way to compute that, or must I work it out in $O(p)$ complexity?
3
votes
3answers
215 views

Comparing $\large 3^{3^{3^3}}$, googol, googolplex

How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$). Thanks much in advance!!!
5
votes
1answer
324 views

Prime decomposition of an integer: methods of determining the prime factors $ p_1, p_2, …, p_r$ and powers $k_1,k_2, …, k_r$

Any integer n can be written in the form $ n = p_1^{k_1}p_2^{k_2} ... p_r^{k_r} $, where the powers $ k_1, k_2, ...,k_r $ are integers and $ p_1, p_2, ..., p_r$ are primes. Now I am interested in ...
15
votes
2answers
338 views

Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime

My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
2
votes
2answers
166 views

Find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$ [duplicate]

I'm trying to find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$, as part of a basic number systems course. I'm rather lost with this question. For $10!$ I tried writing the terms ...
2
votes
1answer
201 views

Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ ...
8
votes
4answers
365 views

Is there a Definite Integral Representation for $n^n$?

The factorial $n!$ has a nice representation as definite integral: $$ n!=\Gamma(n+1)=\int_0^\infty t^{n} e^{-t}\, \mathrm{d}t. \! $$ Is it possible to write down such an integral for $n^n$ as ...
1
vote
0answers
135 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
6
votes
1answer
402 views

Is it known or new? [duplicate]

Possible Duplicate: Starting digits of 2^n While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact. My ...
101
votes
3answers
7k views

Is 2048 the highest power of 2 with all even digits (base ten)?

I have a friend who turned 32 recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being 32 was pretty good since not only is it even, it also has no odd factors. ...
65
votes
4answers
2k views

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
-1
votes
2answers
122 views

Proof of even like powers?

Can someone show me the proof that difference of like even powers of any two numbers is divisible by the sum of the bases?
13
votes
2answers
1k views

How to know if a number is a power of $x$

I couldn't find anything on the Internet which could direct me to the solution of the following problem. I want to know if $n$ can be calculated by $x^y$ where $y\ge 2$ and $x\ge 2$. I tried using ...
5
votes
2answers
348 views

How many positive integer solutions to $a^x+b^x+c^x=abc$?

How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to ...
2
votes
1answer
249 views

Calculating the rightmost digits of Graham's number

Through some miscellaneous reading I have stumbled upon Graham's number and more precisely, a method of calculating the $d$ rightmost digits of the number. The exact method of calculation seems ...
12
votes
3answers
1k views

evaluate the last digit of $7^{7^{7^{7^{7}}}}$

I found this puzzle online. Since I'm not good at number theoretic kind of problems I'm going to propose it in this form. If you have a number $x$, in this case $x=7$, how do you evaluate the last ...
7
votes
1answer
122 views

Proof for the uniqueness of all the combination of the digits of $2^k$, where k>3

A long time ago I found a question on the internet that went a little like this: Suppose that we have $n=2^k$ where $k\gt 3$. If $m$ is another number that is a combination of the digits of $2^k$, ...
1
vote
0answers
76 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...
4
votes
1answer
152 views

Are limits on exponents in moduli possible?

Suppose I show that: $$x^{f(z)/g(z)} = y \pmod{4}$$ is impossible for some given positive integers $x$ and $y$, where, \begin{align*} f(z) &= \phi(4) k_1(z) + 1 \\ &= 2 k_1(z) + 1\\ g(z) ...
31
votes
5answers
2k views

$x^y = y^x$ for integers $x$ and $y$

We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?