A Summer of Maths

$x=\pm 3$ means +3 OR -3 so there is no contradiction.

Edit: didn't see the above post!

Thanks, confused myself. What I don't like that it means, x=3 => x= +/- 3. You know, it's like saying that a green car is green or orange, lol. Need to get used to proper maths, and stop asking silly questions.

Well if you think about, the only other option is AND - which really doesn't make sense!

Indeed, but isn't it x=3 or -3 or both?

Indeed, but isn't it x=3 or -3 or both?

Ok, last question and I will vanish,

If there's a question "solve: x-2=0" is it not technically correct to write x-2=0 => x=2 and finish here. x could be 2, but it doesn't mean that it is, so in order to get it right we must write: x-2=0 <=> x=2?

Wow, it took me a whopping two weeks to get around to writing a solution to your problem...again.
At least I have a legit excuse: I was in Bulgaria participating in the IMC, where I got a Second Prize

In return, I have another very nice group question:
Show that every group of even order contains an element of order two.

Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.

Here is a fun question.....i dont want to give hints as it will give it away

Q. Prove no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

Just found a marvelous proof of this!

(which this post is too narrow to contain...)

Just found a marvelous proof of this!

(which this post is too narrow to contain...)

its only a few lines what the hell did you do

If you want some fun, here are 17 proofs of the infinitude of primes by Tomohiro Yamada.
(You might want to check the references, or find some of the proofs elsewhere.)

Here is a fun question.....i dont want to give hints as it will give it away

Q. Prove no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

Looking at your other posts, I don't think you've managed to say quite what you wanted.

Suppose there are three positive a,b,c such that a^n + b^n = c^n for any integer value of n greater than two.
then
c^4+2a^2b^2=(a^2+b^2)^2
=c^4
so either a or b is zero. contradiction

I like this one.

Let $a$ and $b$ be non-zero relatively prime integers. Show that $\displaystyle \frac{1}{ab}$ can be written as

$\displaystyle \frac{1}{ab} = \frac{x}{a} + \frac{y}{b}$

for some integers $x$ and $y$.

Hint:

Here is a nice result, with a nice proof, that is used to prove the above.

Lemma. The non-zero values, for $x \in \mathbb{N}$, of the non-constant polynomial

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0, \ \ \ a_i \in \mathbb{Z}$

are divisible by infinitely many primes.

It looks good, but I have to admit that I needed Idris Mercer's note.

I like things involving arithmetic progressions in general. They are not as innocent as they appear to be.
The other day, I found an algebraic proof for a special case of Dirichlet's theorem ($b = 1$ in this link).



Here is a nice result, with a nice proof, that is used to prove the above.

Lemma. The non-zero values, for $x \in \mathbb{N}$, of the non-constant polynomial

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0, \ \ \ a_i \in \mathbb{Z}$

are divisible by infinitely many primes.


Does anyone believe it? Or find it obvious?

I can give the proof or consecutive hints, depending on whether people are interested.