3
votes
1answer
37 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
4
votes
4answers
348 views

Is the reasoning/algebra for my proof correct? (musical tuning theory proof)

This isn't for a class, I was just wondering if I would be able to work out a proof for something like this myself for fun, and wanted to verify that my methods are correct. Basically, what I'm trying ...
4
votes
1answer
45 views

Proof verification: Let $a$ be an irrational number and $r$ be a nonzero rational number. If $s$ is a rational number then $ar$ + $s$ is irrational

I have to prove the following: Prove: Let $a$ be an irrational number and $r$ be a nonzero rational number. If $s$ is a rational number then $ar$ + $s$ is irrational So, I decided to do a proof ...
0
votes
1answer
25 views

Proof - Fundamental Theorem of Arithmetic using Euclid's Lemma

Let $n \in Z > 1$. Then the expression for $n$ as the product of $\ge 1$ primes is unique, up to the order in which they appear. From Proofwiki. Suppose $n$ has two prime factorizations: ...
0
votes
0answers
14 views

Are these solutions to Linear Diophantine Equations too? Where'd they hail from?

(1) Can't the signs - I colored them in red - of x and y be switched? Aren't $x = x_0 - bn/d$ and $y = y_0 + an/d$ also solutions? They satisfy $ax + by = c$? (2) How can I remember these ...
5
votes
0answers
35 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
2
votes
0answers
38 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
17
votes
3answers
664 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
1
vote
1answer
20 views

Is this divisibility proof by induction correct/sufficient?

To show: 13 | $4^{2n+1}+3^{n+2}$ I used induction beginning successfully with n=0 (or n=1), then making the step to n+1: An x exists so that $13x = 4^{2n+3}+3^{n+3}$ $13x = 16*4^{2n+1}+3*3^{n+2}$ ...
0
votes
0answers
57 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
0
votes
2answers
43 views

How do $x^{p}+y^{p}=z^{p}$ and $x\equiv y \pmod{p}$ together imply $x\equiv -z \pmod{p}$?

I am reading Marcus' Number Fields and I have been a little stuck following his argument in page $4$ (where he is sketching an argument for Case 1 of Fermat's Last Theorem for primes $p$ for which ...
0
votes
1answer
45 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
1
vote
3answers
84 views

Divisibility test for $4$

Claim: A number is divisible by $4$ if and only if the number formed by the last two digits is divisible by $4$. Here's where I've gotten so far. Let $x$ be an $(n+1)$-digit number. So $x= ...
2
votes
2answers
90 views

Intuition — If $k \in \mathbb{Z}$ and $n \ge 2$, then the n$^{th}$ root of k is either an integer or irrational.

Origin — Elementary Number Theory — Jones — p25 — Exercise 2.4 (1) How do you prefigure the answer? Proofwiki start after 'auguring' the answer. (2) What's the intuition? This answer delineates ...
2
votes
1answer
100 views

Modified Euclid's proof of infinite primes

Q. Alternate the proof for Euclid's infinite number of primes to show there are infinitely prime numbers of the form $6n-1$ where n is an integer. my attempt, suppose by contradiction there are ...
0
votes
0answers
55 views

Question on a “proof”

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. Define the abundancy index $I$ as $I(x) = \sigma(x)/x$. Suppose that we have the condition $$I(a^2)I(b^c) = 2.$$ Let $a$ and $c$ ...
2
votes
4answers
109 views

Stuck while trying to prove $2k^3 \geq (k + 1)^3$…

how can I prove the following: $2k^3 \geq (k + 1)^3$ This is the final part of the elaborate proof for $2^n > n^3 $ give $ n \geq 10$ I have used induction and end up with: $ 2^{K+1} > 2k^3 $ ...
3
votes
3answers
117 views

Is my intuition of “If $p \mid ab$ then $p \mid a$ or $p \mid b$” correct?

I'm studying number theory and I was given this Theorem to look at: If $p \mid ab$ then $p \mid a$ or $p \mid b$ I had the following intuition for the problem or a proof of sorts if you will. ...
7
votes
1answer
164 views

Proof that Fibonacci Sequence modulo m is periodic?

It's well known that the Fibonacci sequence modulo m (where m is any integer) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more complicated. ...
5
votes
2answers
98 views

Proof - There're infinitely many primes of the form 3k + 2 — origin of $3q_1..q_n + 2$

Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6 To instigate a contradiction, postulate $q_1,q_2,\dots,q_n$ as all the primes $\neq 2 (=$ the only even prime) of the form $3k+2$. ...
2
votes
4answers
164 views

Determination of the last three digits of $2014^{2014}$

May I know if my proof is correct? Consider $x$ such that $2014^{2014} \equiv x \pmod{1000}.$ By Euler's theorem, $2014^{\ \psi(1000)} =2014^{\ \psi(2^3)\psi(5^3)}=2014^{400}\equiv 1 \pmod{1000}$. ...
2
votes
2answers
121 views

Determination of the last two digits of $777^{777}$

May I know if my proof is correct? Thank you. This is equivalent to finding $x$ such that $777^{777} \equiv x \pmod{100}.$ By Euler's theorem, $777^{\ \psi(100)} =777^{\ 40}\equiv 1 \pmod{100}$. ...
8
votes
1answer
93 views

Proof by induction that $\alpha^n + \beta^n \in \mathbb Z$

Let $\alpha, \beta \in \mathbb C$ such $\alpha + \beta \in \mathbb Z$ and $\alpha \beta = j \in \mathbb Z$. Prove that for all $n \in \mathbb N,\alpha^n + \beta^n \in \mathbb Z$ Is this proof ...
2
votes
1answer
27 views

Salvage of a given propostion

Consider the statement: For all integers $r$, $s$, and $a$, and natural numbers $m$, if $ra \equiv sa \pmod m$ then $r\equiv s \pmod m$. I have found this statement to be false by the counterexample ...
1
vote
0answers
86 views

Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$

All the units are satisfied Pell's equation $a^2-2b^2=\pm1$ for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$. Here is my proof: Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies ...
0
votes
2answers
47 views

6 is a unique number $n$ such that $n-LD(n)^2 = 2$

Let $LD(n)$ be the lowest divisor of $n$ larger than $1$. Let's find all numbers $n$ such that $n-LD(n)^2 = 2$. If $n$ is even then $LD(n) = 2$ and $LD(n)^2 = 4$. Plugging in we get $n-4=2$, so $n=6$. ...
0
votes
0answers
23 views

smallest divisor problem - proof verification

Let LD(n) be smallest divisor of n greater than 1. Let's consider the expression n-LD(n)^2. It can be proven that this expression is positive only for composite numbers. Also 6-LD(6)^2 = 2 The ...
1
vote
0answers
47 views

Could Someone Just Verify This Proof for Me? (Euler's Theorem)

I came up with this proof for my number theory class. Is it valid? Proposition: $u\in U_m \Rightarrow u^{\varphi(m)}=1$ (Where $U_m$ is the multiplicative group of integers modulo $m$) Attempted ...
1
vote
1answer
50 views

Proving (by using Zorn's lemma) that every nonempty set contains a maximal ideal

I am trying to prove the following exercise: Let $X \neq \emptyset$. Prove, (by using Zorn's Lemma) that there exists a maximal ideal in $P(X)$. Proof: Take $\mathcal{J}$ to be the set of all ideals ...
1
vote
1answer
589 views

Prove the proposition: there are infinitely many primes of the form 4k + 3, where k ≥ 0 is an integer

Proposition 2. there are infinitely many primes of the form 4k + 3, where k ≥ 0 is an integer. (a) Let n ∈ N. Suppose q1,q2,...,qn are positive integers such that for all 1 ≤ i ≤ n, each qi = 4ki + ...
0
votes
1answer
171 views

Do I have this right? Are these conclusions valid in this isomorphic view of $\Bbb{R}$?

Let $F = (\Bbb{R}, \oplus_d, \cdot)$ be the field with usual $\cdot$, and $\oplus_d$ is defined as $a \oplus b = (\sqrt[d]{a} + \sqrt[d]{b})^d$. This field is isomorphic to usual $\Bbb{R}$ structure ...
0
votes
1answer
49 views

set theory, show sets are not of equal cardinality - check my proof

question from exam in set theory: let $M$ be the set of all real numbers x that satisfy: $cx^2+bx+a=0$ where $a,b,c \in Z$ (Meaning they are integers) and $c$ is not $0$. We will define $K = \{sm+t ...
1
vote
3answers
70 views

Prove by induction that $3\mid (n^3 - n)$

I'm having an argument with my professor whether my exam was right or not. Before I sign a formal complain to get a review on my exam, I'd like to be sure it's correct. My answer: Proof by induction: ...
1
vote
3answers
270 views

Prove directly, by contradiction, or contraposition? If the product of two integers is even, at least one of them must be even.

Hello I am having some trouble trying to know which way I should go as for this proof: ( we are suppose to use either direct, contradiction or contraposition) Prove or disprove: If the product of two ...
4
votes
0answers
96 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...
0
votes
0answers
63 views

Elementary proof of Beal's conjecture

If $x^a + y^b = z^c$ and $$ 1 = r_0 x^a + s_0 y^b \\ 1 = r_1 x^a + s_1 z^c \\ 1 = r_2 y^b + s_2 z^c $$ i.e. all the $\gcd$'s between $x,y,z$ are $1$. Then we can write $$ 1 = r_0 x^a + s_0 y^b ...
1
vote
0answers
67 views

Is this transformation of Beal's conjecture valid?

Beal's conjecture is: If $$ x^a + y^b = z^c \ \ \ \ (1) $$ where $a,b,c, x,y,z$ are positive integers with $a,b,c \gt 2,$ then $x,y,z $ have a common prime factor. (copied from Wikipedia) ...
3
votes
3answers
106 views

last two digits of $9^{1500}$ (Dummit Foote -Abstract Algebra preliminaries $0.3.5$)

Question is to find last two digits of $9^{1500}$ (No Euler totient theorem please) What i have done so far is : $9^2\equiv 81\text{mod} 100$ $9^4 \equiv 61\text{mod} 100$ $9^8\equiv ...
1
vote
2answers
269 views

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n [duplicate]

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n Hi everyone, for the proof to the above question, Can I assume that since $(a, b) = ...
1
vote
1answer
171 views

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$ Hi everyone, I would like to know if my assumption is justified for answering the above question. Any ...
3
votes
0answers
194 views

Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$

Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$. Hi there, I want to know if the following proof I have is strong enough, or if I'm making false assumptions :|. ...
1
vote
1answer
32 views

Help with understanding this proof (I think it's Hensels Lifting?)

I am reading a proof that shows that if $a$ is a quadratic residue modulo $p^k$, where $p$ is a prime > $2$ and $k$ is a positive integer, then it is also a quadratic residue modulo $p^{k+1}$. Here ...
2
votes
2answers
52 views

Proof of statement: If $a\mid b$ and $a\mid c$, then $a \mid b+c$

Statement: If $a$ divides both $b$ and $c$, then $a$ divides $b+c$ Proof: Assume that $a$ does not divide $b+c$. Then there is no integer $k$ such that $ak=b+c$. However, $a$ divides $b$, so $am=b$ ...
2
votes
5answers
133 views

How to show that $(3k+2,5k+3)=1$ for all $k\in\mathbb{Z}$

I think I'm on the right track, but I can only figure out how to prove for a specific $k$ of my choosing... I don't know how to generalize it for all $k$: Assume $(3k+2,5k+3)=1$. Therefore, there ...
2
votes
2answers
316 views

Prove that the Diophantine equation $ax+by+cz=e$ has a solution if and only if $(a,b,c)\mid e$.

I have an intuitive idea about how this is going to work, but I don't know if I'm writing it properly using proper math language and theorems. I am most uncomfortable with the second half of the proof ...
3
votes
1answer
139 views

Can powers of primes be perfect numbers?

I need to prove the following, though I'm not 100% certain I understand the definition of a perfect number. Prove that no perfect number is a power of a prime. First of all, I'm assuming that ...
2
votes
2answers
67 views

Number theory proof from AoPS

http://www.artofproblemsolving.com/Resources/articles.php?page=htw.readers In the above link, he gives a problem, namely Let $S(n)$ be the sum of the digits of $n$. Find ...
1
vote
1answer
76 views

Proof Checking and input: Generators of $\mathbb{Z}_{pq}$

I'm self-studying abstract algebra (slowly but surely), and I have a question about my answer to the following prompt: Problem statement: Show that there are $(q-1)(p-1)$ generators of the group ...
1
vote
5answers
67 views

Techniques of Proof

Prove If n is an integer, then $n^2+n^3$ is an even number. I don't know if I'm just reiterating what I'm asked to prove or if my ideas are actually proving the statement. If $n^2+n^3$ is an even ...
5
votes
3answers
1k views

Strong Induction Proof: Fibonacci number even if and only if 3 divides index

The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$ Proof by Strong Induction : ...