0
votes
2answers
37 views

Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple ...
1
vote
1answer
32 views

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

I computed $\dfrac{\sigma_1(n)}{n}$ and $\sigma_{-1}(n)$ on a good hundred values of $n$, and they seem to always match. For example: $\dfrac{\sigma_1(6)}{6} = \dfrac{1 + 2 + 3 + 6}{6} = ...
0
votes
2answers
86 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
1
vote
1answer
127 views

Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.

I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers. A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers ...
3
votes
2answers
87 views

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
2
votes
1answer
24 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : *$\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$ by definition we have ...
0
votes
2answers
14 views

Proof that in any base $b$, the result of multipling two numbers of $k$ digits, doesn't recuire more than $2k$ digits

The proof that I came up whit is: Let, $c$ be $b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k r_k$ and $d = b^0 r_0'+ b^1 r_1'+b^2 r_2'+...+b^k r_k'$ then multipling both: $$(b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k ...
0
votes
4answers
69 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
5
votes
1answer
95 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
5
votes
2answers
199 views

Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$?

If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or ...
1
vote
0answers
92 views

Proof Synopsis of Fermat's Last Theorem

I'm taking a introduction to higher math course now (mostly number theory) and our professor wants us to include two sentence proof synopses with our longer proofs. This got me thinking, What is a ...
1
vote
2answers
52 views

Finding a real number c for polynomial (proof)

The question is to find a real number c for which $ x\ge c%+$ implies $$x^4-4x^3+7x-9 \ge1000$$. I was given the hint that $x>10$, then $4x^3<0.4x^4$, so $x^4-4x^3>0.6x^4$. Problem is, I'm ...
1
vote
1answer
47 views

Math Proof Question similar to reverse triangle inequality

Prove that for any real three numbers x,y,z, $$ |x-y||z| \le |y-z||x| + |z-x||y|$$ I am way overthinking this, there must be an easier solution to this. Any thoughts?
3
votes
1answer
102 views

Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
1
vote
2answers
144 views

Primitive Roots Proofs

I am stuck on how to prove these two questions: (1) Let r be a primitive root of the prime $p$ with $p$ congruent to $1$ modulo $4$. Show that $-r$ is also a primitive root. (2) Let n be a positive ...
1
vote
3answers
34 views

Let $m$ and $n$ be integers in the ring of integers. Show that if $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$

Hello everyone working on the problem in the title...it's an if and only if proof so two directions to show start with the fact $n\mathbb Z$ is a subset of $m\mathbb Z$ and want to show $n=mx$ for ...
1
vote
1answer
73 views

How do I prove: Let n ∈ N+. Let m ∈ N+. m<n. Prove that n⊥m ⇒ (n−m)⊥ m.

I don't even know how to start, any help would be greatly appreciated!
0
votes
4answers
2k views

Proof of the statement “The product of 4 consecutive integers can be expressed in the form 8k for some integer k”

I am slowly diving into simple number theory and learning how to craft direct proofs. I needed to proof the statement "The product of 4 consecutive integers can be expressed in the form 8k for some ...
2
votes
3answers
68 views

Proofs and Number theory

I am needing help proving the following: For any integer $n$, $n^2$ + 5 is not divisible by $4$ I am aware that an integer $x$ is divisible by integer $y$ if there exists integer $k$ such that ...
2
votes
1answer
86 views

how to show associativity of multiplication for not just 3 operands but for n operands

ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
3
votes
4answers
786 views

Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$

I want to prove by induction that $a^n \mid b^n$ implies that $a \mid b$ holds for all integers $n\geq 1$. clearly for $n=1$ this is true, since if $a \mid b$, then $a \mid b$. Suppose this is true ...
1
vote
1answer
156 views

Are there any errors in my proof that only perfect squares have rational square roots?

This is a very simple proof, but I know that proofs which are this simple can often have some erroneous assumptions. Is mine okay? Argument: With the exception of perfect squares, there are no ...
8
votes
3answers
357 views

Proof of Wolstenholme's theorem.?

According to the theorem : $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$ And we have to prove that $r= 0 \pmod{p^2}$. (Given $ p>3$, ...
0
votes
4answers
48 views

Prove that if $a$ is a prime, $b_i \in \mathbb{Z}_+$ and $a | \prod_{i = 1}^{n} b_i$ then $a | b_i$ for some $b_i$

A key property of the integers is that: if $\gcd(a,b) = 1$ and $a |bc$, then $a|c$. Use this property to prove that: if $a \in\mathbb{Z}_+$ is prime and $b_i \in \mathbb{Z}_+$ for $1 \leq i \leq n$ ...
4
votes
5answers
258 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
6
votes
2answers
163 views

How to show that: $\gcd\left( {a^n-b^n \over a-b} ,a-b\right)=\gcd(n d^{n-1},a-b )$

How to show that: $$ \gcd\bigg( {a^n-b^n \over a-b} ,a-b\bigg )=\gcd(n d^{n-1},a-b ) $$ $a,b\in \mathbb Z$ where $d=\gcd(a,b)$? $\gcd$ is the greatest common divisor.
0
votes
1answer
143 views

Prove that $p\mid \binom{p}{k},\ 0< k< p$

Prove that: $$p \,\,\left|\, {p \choose k} \right., \quad 0< k \lt p$$ if $p$ is prime. how to prove that with direct proof?
0
votes
3answers
237 views

Prove that for any integer $k \ne 0$, $gcd(k, k+1) = 1$

I'm learning to do proofs, and I'm a bit stuck on this one. The question asks to prove for any positive integer $k \ne 0$, $gcd(k, k+1) = 1$. First I tried: $gcd(k,k+1) = 1 = kx + (k+1)y$ : But I ...
1
vote
1answer
132 views

A probabilistic method

I am trying to study for a exam and i found a assignmet, that i cant solve. Consider a board of $n$ x $n$ cells, where $n = 2k, k≥2$. Each of the numbers from $S = \{1,...,\frac{n^2}{2}\}$ is written ...
18
votes
10answers
3k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
1
vote
3answers
281 views

Advice for a newly minted math major

I just recently decided to go back to school. I had previously majored in piano performance and dropped out to work as a software engineer. In high school, I did pretty well at math, I got a 5 on ...
1
vote
4answers
7k views

Prove the square root of 7 is irrational using the Division Algorithm and case reasoning

I proved this previously using proof by contradiction like so: I am not seeing where to start to prove it using the Quotient Remainder theorem or case reasoning however. Can anyone see the best way ...
3
votes
1answer
296 views

Is this a good proof of Wilson's theorem? — ($(n-1)!+1 \equiv_n 0$ iff n is prime)

Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime. To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p: ...
2
votes
1answer
356 views

Proof of max product of partitions of n

For $n \in \mathbb{Z} : n \geq 1$ $ f(n) = \displaystyle\max_{\substack{ x_1+\dotsm+x_k = n\\ x_i\in\mathbb{Z}^{+} }} x_1 x_2 \dotsm x_k $ $$ f(n) = \begin{cases} 1 & \text{if ...