Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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3
votes
2answers
58 views

Is the last digit of this number :$ {{4^4}^n}+1 $ always $7 $ for $n>1$ and could this be prime?

Some computations in wolfram alpha for $n=2,3,4,5 ,6$ showed that the last digit of this number $ {{4^4}^n}+1 $ for $n>1$ always $7$ . My question here :How do I know if it's last digit always ...
17
votes
3answers
898 views

Prove that the number 14641 is the fourth power of an integer in any base greater than 6?

Prove that the number $14641$ is the fourth power of an integer in any base greater than $6$? I understand how to work it out, because I think you do $$14641\ (\text{base }a > 6) = ...
2
votes
0answers
22 views

How do i show this :$\lim_{k\to\infty} \frac{\sigma_{2k+1}(n)}{\sigma_{2k-1}(n)}=n²$ if it is true?

I run some computation in wolfram alpha I find for many fixed values of $n$ and for an arbitrary integer $k$ the ratio : $\frac{\sigma_{2k+1}(n)}{\sigma_{2k-1}(n)}$ close to $n²$ . My question here ...
0
votes
3answers
32 views

Divisibility of integer numbers

If we have two integers $a$ and $b$ such that $a = \frac{5b}{6}$, is $a$ divisible by $5$? If so, why is that? I don't see it.
0
votes
5answers
51 views

Is the following proof correct for $n(n+1)(n+5) = 3X$

The objective is to prove that $n(n+1)(n+5)$ is a multiple of 3. I took the following simplistic route $$n(n+1)(n+5) = 3X$$ $$n(n+1)(n+5)\frac{n+2}{n}\frac{n+6}{n+5} = ...
0
votes
1answer
27 views

Under what conditions is the implication $I(x) < I(y) \implies x < y$ true?

Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $$I(x) = \frac{\sigma(x)}{x}.$$ My question is: Under what conditions on $x$ and $y$ is the implication ...
1
vote
2answers
40 views

Finding error in an incorrect proof

Statement: If $a$, $b$ and $b'$ are integers and $a>b>b'>0$, then the remainder when $a$ is divided by $b$ is less than the remainder when a is divided by $b'$. Proof: Assume $a,b, b'$ are ...
0
votes
1answer
51 views

What do I gain if a number is Quadratic residues?

If in a random question, I see: $1)\:\: x^2 = a\pmod p$, $p$ is a prime. $2)\:\: a$ is quadratic residues $(QR)$. How can I conclude that $(a, p)=1$ ? What is missing to determine that? I learned in ...
3
votes
0answers
55 views

Fibonacci numbers properties

I've verified that $F_{41} \mod F_{32}= F_{23}$ where $41-32=9$ and $32-9=23$. I suppose these facts are correlated. Is there a simple way to show how? Simpler question: how can I justify that ...
0
votes
1answer
61 views

How to express the function $\mathbb{N} \to \mathbb{N}\times \mathbb{N}$ as a mathematical statement?

I am not that good at creating proofs, but I am decent at coding and was able to come up with this simple "program" that makes up the function I desire: ...
0
votes
2answers
46 views

How to find pythagorean triples and n-tuples

A pythagorean triple is any set of three positive integers $(a,b,c)$ where $a^2 + b^2 = c^2$ I'm wondering, is there a formula to find all pythagorean triples, and can it be generalized to ...
0
votes
1answer
56 views

Find sum of two primes if their difference is equal to $3{ n }^{ 2 }-5n-1$

The difference between two prime numbers is equal to $3{ n }^{ 2 }-5n-1$. By using $n$, find the sum of them, where $n \in \mathbb{N}$. I didn't have any idea about how can I start to solve ...
6
votes
1answer
93 views

Are there any two squares with a distance of a power of 2 between them?

I was working with Pythagorean triples and found a formula to generate Pythagorean triples. I am pretty sure every Pythagorean triple falls under my formula. This formula is: If $a$, $x$, and $n$ are ...
0
votes
1answer
35 views

Prove that there are as many quadratic residues mod p as there are quadratic non-residues mod p [duplicate]

I have found various proofs of this question usig primitive roots, but I want to prove it without using primitive roots! Here is my question again: Let $p$ be prime. Prove that there are the same ...
0
votes
3answers
32 views

Linear Combination of Roots of Unity

Let $\omega_n$ be a primitive $n$the root of unity and $\lambda_k$ be natural numbers. Does $\sum_{k=1}^{n} \lambda_k w_n^k =0$ imply $\lambda_1 = \lambda_2 = ... = \lambda_n $? I am aware of ...
4
votes
1answer
37 views

Sequence of digits $7143$ appears somewhere after the comma

Let $m,n$ be positive integers such that in the decimal representation of $\frac{m}{n}$, the sequence of digits $7143$ appears somewhere after the comma. Prove that $n>1250$. For small values of ...
1
vote
1answer
100 views

Find the last $2$ digits of $7^{7^{7^{10217}}}$ [duplicate]

Find the last $2$ digits of $$\large7^{7^{7^{10217}}}$$ So far I have: $17\cdot 601=10217$ and $7=7 \pmod{10}$. Any help greatly appreciated.
0
votes
3answers
51 views

Proving that if $a|b$ and $b|a$, then $a = \pm b$ for $a,b$ as nonzero integers?

Can someone walk me through how to do a proof of the following? Let $a$ and $b$ be nonzero integers. Use a direct proof to show that if $a|b$ and $b|a$, then $a= \pm b$. So I know $a,b \neq 0$ in ...
2
votes
2answers
15 views

Successors of perfect squares are not divisible by predecessors of double even numbers

Is there a proof for: successors of perfect squares are not divisible by ( not multiple of ) predecessors of double even numbers, i.e., $n^2+1$ not divisible by $4k-1$, or $3+4k$?
1
vote
1answer
61 views

The ring $R_{p}$

Let p be a prime number in $\mathbb{Z}$. Let $R_{p}$ be the ring $R_{p} : = \{x \in \mathbb{Q} : ord_{p}(x) \geq 0\}$. Show that x is a unit of R if and only if $ord_{p}(x)= 0$. I'm not sure how to ...
2
votes
2answers
59 views

A triple of pythagorean triples with an extra property

I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align} (x-z)^2+y^2 &\text{ is a perfect square,}\\ x^2+y^2 &\text{ is a perfect ...
0
votes
0answers
39 views

Show that $504 \mid n^9 − n^3 $ for any integer $n$ [duplicate]

Not sure how to start this. I know that $504 =2 \times 2 \times2 \times 3 \times 3 \times 7$.
2
votes
0answers
22 views

Euler totient divisor sum [duplicate]

To prove this identity, $$\sum_{d \mid n}\phi(d)= n \qquad \text{for} \, n=1,2,3,\ldots$$ where $\phi (n)$ is the Eulers totient function, I tried this by breaking it into two parts, n is either an ...
0
votes
1answer
41 views

Which two integers produce random infinite sequence when the largest divided by the smallest?

Which two integers produce random infinite sequence when the larger one is divided by the smaller one? For instance, $\frac{920}{33}= 27.8787878787...$, is not a random sequence.
0
votes
0answers
39 views

On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

(Note: I have edited this question to conform to the further details added in the cross-post to MO.) Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if ...
0
votes
2answers
52 views

Solving modulo equations with one variable

Given the following equation: $$10 = 4^x \pmod {18}$$ How can one know what are the correct values for $x$ ?
1
vote
1answer
13 views

When does:$(p^y+1 )\bmod (p^x+1)=0 $ if $(y,x)=1$ and $p $ is a prime number?

I'm interesting to look the solution of this equation :$$(p^y+1 )\bmod (p^x+1)=0 $$ at a least to see an example of the two coprime $y, x$ for which $(p^y+1 )\bmod (p^x+1)=0 $ but i don't succed , ...
0
votes
0answers
47 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z*}_{p},\cdot)$ with the ...
0
votes
1answer
14 views

Finding the number of multiples in an interval [1, x] using the floor function

Let $m$ be a positive integer. Show that for every real number $x \geq 1$, the number of multiples of $m$ in the interval $[1,x]$ is $\left \lfloor \frac{x}{m} \right \rfloor$. I am taking an ...
2
votes
3answers
69 views

Some hints for “If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$”

I tried to prove this question by first considering the possible last digit of $p$ when $p=n^2+5$, but that reasoning got me nowhere. Then I tried to prove it by contrapositive, and however I just ...
0
votes
1answer
26 views

Can you square (or exponentiate to any power) both sides of a modular equation?

If $ a \equiv b \;\;(mod\;p) $ then by definition this means $p | (a - b)$. Now for $n = 2$ we would have $ a^2 \equiv b^2 \;\;(mod\;p) $ or $p | a^2 - b^2$ and $p | (a - b)(a + b) $. Now this will ...
3
votes
1answer
47 views

2 is a square modulo $p$ if and only if $p \equiv \pm 1 \pmod 8$

2 is a square modulo $p$ if and only if $p \equiv \pm 1 \pmod 8$ The indication for this exercise was to consider $\alpha$ such that $\alpha^8 = 1$ ($\alpha$ is eigth root of the unity) in a ...
2
votes
1answer
41 views

Density of numbers with exactly $n$ distinct prime factors in $\mathbb{N}$

It is quite well known that the density of the primes in $\mathbb{N}$ is $0$, that is, $$\lim_{n\to\infty}\frac{|\{p\mid p\leq n, p \text{ prime}\}|}{|\mathbb{N}_{\leq n}|}=0$$ It is less well-known, ...
3
votes
1answer
90 views

Let $x$ be a real number. Prove the existence of a unique integer $a$ such that $a \leq x < a+1$

Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$ ...
0
votes
2answers
48 views

Minimum number of moves to even out a row of brick piles

Consider a row of $15$ piles of bricks. There is a total of 75 bricks, all identical. The number of bricks per pile varies across the piles. For instance, the distribution of bricks per pile might be ...
0
votes
1answer
17 views

Working out leftovers from grouping

Mike has 2015 matches and decides to put them in a triangular pattern. He starts with three at the top,then five,then seven,and so on. In the end ,how many matches does he have left over. When I ...
1
vote
2answers
63 views

$X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers

I am trying to prove that $X^4 - 4Y^4 = -Z^2$ has no solutions in non zero integers. I know there are similar questions on MS, but that minus signs before the $Z$ gives me a hard time. For the ...
0
votes
1answer
80 views

Any counter example for this claim?

I would like to proof or disproof this claim ,but i don't have enough information about divisor function structure . Claim : for any positive integer $x, y ,n $ such that :$x\neq y$ and ...
3
votes
0answers
36 views

Fill unit square in Euclid plane [closed]

Assume $A,B$ are two subset of $\mathbb{Z^2}$. In addition, $A$ is finite. Satisfies: i) For all $a_1,a_2 \in A$ and $b_1,b_2 \in B$ , $a_1+b_1=a_2+b_2$ implies $a_1=a_2$ and $b_1=b_2$ ii) ...
1
vote
1answer
26 views

Divisibility of a certain power sum

Does there exist any even integers $m,n$ such that $m$ divides $1^n + 2^n + \cdots +(8m+2)^n$ ? After my initial attempt below that i'm not so sure of, i feel that Bernoulli polynomials might be ...
1
vote
0answers
39 views

Elementary divisibility problem.

I was doing an elementary number theory exercise in a certain text, and i came across this problem: Is there an even integer $n$ such that $1^n + 2^n + \cdots +m^n$ divides $(m+1)^{n}(m^n - 2^n)$, ...
-5
votes
1answer
136 views

Can I belive that : $e^{e^{e^{e^{\cdots}}}}$ is $\infty$? [closed]

Definetly this number : $e^{e^{e^{e^{\cdots}}}}$ is not an integer this implies that is not prime number or perfect number , now i would like to know really what is the nature of this number ...
6
votes
5answers
76 views

Why can we convert a base $9$ number to a base $3$ number by simply converting each base $9$ digit into two base $3$ digits?

Why can we convert a base $9$ number to a base $3$ number by simply converting each base $9$ digit into two base $3$ digits ? For example $813_9$ can be converted directly to base $3$ by ...
4
votes
1answer
93 views

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$? [closed]

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$?
0
votes
1answer
29 views

Calculate the sum under gcd

I have a question: How can I calculate $$\sum_{i=1\atop \gcd(i,75)=1}^{75} i=?$$
5
votes
3answers
133 views

Is $x^2+x+1$ divisible by $101$, if $x\in\mathbb Z$?

Prove $x^2+x+1$ isn't divisible by $101$, for any $x\in\mathbb Z$? I think the way of solving the problem it by using "Fermat's Little Theorem".
8
votes
3answers
256 views

Finding the solution of a congruence.

Solve the congruence $$4x\equiv16\mod{26}.$$ How do I find the solution to this? I have tried by the euclidean algorithm but the gcd is not $1$ so it doesn't work. $$\begin{align} ...
0
votes
1answer
64 views

Product of two distinct odd primes is not carmichael

Let $p,q$ be distinct odd primes. Define $N=pq$. Then $N$ is not a carmichael number. Proof: Suppose $N$ is carmichael. By the Chinese remainder theorem we can find a primitive root ...
2
votes
0answers
52 views

The smallest number that can be written as the sum of two or more consecutive integers in exactly 1000 ways

What is the smallest number that can be written as the sum of two or more consecutive integers in exactly 1000 ways? I was reading this question from Link . I have understood the theory behind it. ...
0
votes
1answer
36 views

Exercise 3 on page 5 and exercise 7 on page 6 in Koblitz's Introduction to modular forms.

I want to prove $1$ cannot be a congruent number, by using the fact that if it were congruent then the equation $x^4-y^4=u^2$ would have a solution in integers with $u$ being odd. I proved this last ...