1
vote
0answers
47 views

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ primes. What are the first values of $U(n)$?

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ prime numbers (except for the first prime number: $2$). What are the first values of $U(n)$ up to ...
2
votes
3answers
159 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
6
votes
3answers
134 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
2
votes
1answer
115 views

Solve for x when $2222^{5555} + 5555^{2222} \equiv x \pmod{7}$ [duplicate]

I need to find the remainder when $2222^{5555} + 5555^{2222}$ is divided by $7$. I'm thinking that Fermat's Little Theorem might help. Any suggestions?
1
vote
1answer
58 views

Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = ...
0
votes
0answers
34 views

Find $a,b,c \ge 2$ and $p,q$ odd primes where $p^a - 1 = c*q^b$

I've been recently thinking about finding primes $p,q$ where the power of one divides the power of the other when subtracted by $1$. For example, if we remove the requirement that $p,q$ be odd ...
4
votes
2answers
129 views

Last digits don't change when exponentiating

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
4
votes
3answers
49 views

Graph of the last digit of $x^n$ - why is it symmetric when $n$ is even, and not when $n$ is odd?

I have discovered this fact: "The graph of the last digit of $x^n$ (where $x$ is positive) is asymmetrical if $n$ is odd, and symmetrical if $n$ is even." What is the logic behind this? For ...
2
votes
2answers
106 views

Is it possible to solve $i^2+i+1\equiv 0\pmod{2^p-1}$ in general?

While looking at the Mersenne numbers (for prime $p$, the number $2^p-1$), I noticed that only certain of them had any solution to the modular equation $i^2+i+1\equiv 0\pmod{2^p-1}$, e.g., ...
1
vote
4answers
153 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
1
vote
2answers
60 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
2
votes
0answers
82 views

Sums of powers.

Here's the problem: Show that $19^{19}$ is not the sum of a fourth power and a positive or negative cube. I'm just not really sure how to start approaching this problem. Does anybody have any ...
0
votes
0answers
62 views

What is the number theory behind this?

I am given $3^{1000}$ and asked to find, in base $2$, now many digits it takes to represent this number. According to Wolfram, it is $1585$, but I don't know why. I understand that $2^n$ would be ...
0
votes
2answers
73 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
2
votes
3answers
199 views

How to simplify $42^{25\sqrt{25^{25}}}$?

Am a student preparing for GRE, I have no clue to solve this am attaching the screenshot of question: I need you give me a short cut or tip to deal such problems...
4
votes
1answer
88 views

Does there exist a constant $\sqrt[4]{2} < A < \sqrt2$ such that $\lfloor A^{2^n} \rfloor$ is a practical number for all $n \in \Bbb Z^+$?

Does there exist a constant $\sqrt[4]{2} < A < \sqrt2$ such that $\lfloor A^{2^n} \rfloor$ is a practical number for all $n \in \Bbb Z^+$? I know we can exclude the range ...
1
vote
3answers
263 views

Finding unit's digit in exponentiation

Could someone please explain to me how to find the unit's digit in the following expression: $$7^{95} - 3^{58}$$
2
votes
1answer
168 views

Number of digits and last digit of a number

How can I find the number of digits and the last digit of the number $$\large{2357^{2357^{.^{.^{.^{2357}}}}}}$$ Basically $2357$ to the power of $2357, 2357$ times.
10
votes
1answer
397 views

Is 2201 really the only non-palindromic number whose cube is palindromic?

Hі, Wikipedia states that 2201 is the "only known non-palindromic number whose cube is palindromic", and lists no reference. It is in fact true that $2201^3=10662526601$, which is a palindrome. But ...
2
votes
1answer
69 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. ...
2
votes
1answer
55 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 ...
9
votes
7answers
2k views

Pattern to last three digits of power of $3$?

I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator. I've tried to find a pattern but can not ...
3
votes
1answer
84 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 ...
8
votes
1answer
150 views

Proving a number defined by a sequence is a square number

I found this problem in a math magazine: Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by: $$ x_0 = 0\\ x_1 = 1\\ x_{n+2}+x_{n+1}+2x_{n}=0 $$ Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
102
votes
9answers
4k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
8
votes
3answers
576 views

Is $n^n$ a perfect square or not?

If $n$ is an integer, how do you know whether $n^n$ is a perfect square, without a calculator? The actual question is: "how many integers between $1$ and $100$ inclusive, raised to their own power, ...
4
votes
3answers
834 views

Calculating $17^{14}\mod{71}$ using Fermat's little theorem

Calculate $17^{14} \pmod{71}$ By Fermat's little theorem: $17^{70} \equiv 1 \pmod{71}$ $17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$ And then I don't really know what to do from this point ...
8
votes
1answer
454 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
2
votes
1answer
106 views

Is it possible to know if sums of powers of a number is divisible by another number?

Is there a way to find whether a number (say $A$) formed by summing powers of another number (say $B$) is divisible by another number $C$? $A$ is a number like, for example, $B^1+B^3$. We can use a ...
7
votes
5answers
437 views

Proof that $6^n$ always has a last digit of $6$

Without being proficient in math at all, I have figured out, by looking at series of numbers, that $6$ in the $n$-th power always seems to end with the digit $6$. Anyone here willing to link me to a ...