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# Complicated proofs of simple facts...

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1. Theorem: The sum of the degrees of the vertices of a graph is even.
Proof: The number of graphs with degrees is the coefficient of in the generating function . Now apply Cauchy's Theorem in complex dimensions, giving

where each integral is a simple closed contour enclosing the origin once.
Choosing the circles , we get

Alternatively, choosing the circles , we get

where . Since when is an odd integer, we can add these two integrals to get .

Theorem: is even.
Proof: is the order of the group (see http://groupprops.subwiki.org/wiki/Alternating_group:A5). It is known that is a non-abelian simple group. Therefore is not solvable. But the Feit-Thompson Theorem asserts that every finite group with odd cardinal is solvable, so must be an even number.

Theorem: is irrational for .
Proof: If then , contradicting Fermat's Last Theorem. Unfortunately FLT isn't strong enough to prove the irrationality of though.
Alternative Proof: is an irreducible polynomial, by Eisenstein's Criterion.

Theorem: The harmonic series diverges.
Proof: Assume it is convergent. Then the functions would be dominated by an absolutely integrable function. But

Theorem: There are infinitely many primes.
Proof: is irrational, by Apery's Theorem.

Theorem: All integers can be written as the sum of two squarefree integers.
Proof: We can manually check that the result holds for . Now let be the set of all squarefree integers, except the primes larger than . Then using
(a) the fact that the Schnirelmann density of the set of squarefree integers is - see http://www.jstor.org/pss/2034736, and
(b) some decent estimate on the prime counting function, e.g one of the inequalities at https://en.wikipedia.org/wiki/Prime-...n#Inequalities
we have that the Schnirelmann density of is greater than . By Mann's Theorem, we now have that every positive integer can be written as the sum of at most elements of . In particular, every prime number, and every non-squarefree integer, can be written as the sum of exactly elements of . Now all that remains is to prove the theorem for composite squarefree integers, which can be done as follows: , where is the smallest prime dividing and are squarefree integers.
2. Nvm
3. Oh this wasn't a question sorry my bad. Ignore the above as you probably know it.

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4. Euclid's proof of infinitely many primes is much more elegant...
5. I'm stuck boys how 2 prv 5!/2 is even

I have so far

5!=5x4x3x2x1=120 so 5!/2=120/2 but I don't know how to finish it.

Send help....

Hapax you compensating for anything with these proofs I wonder?
6. (Original post by cliveb2016)
I'm stuck boys how 2 prv 5!/2 is even

I have so far

5!=5x4x3x2x1=120 so 5!/2=120/2 but I don't know how to finish it.

Send help....

Hapax you compensating for anything with these proofs I wonder?
I think the questn is rong. Coz 1/2 is 0.5 which isnt a whole number so the proof is brocken.
7. Proof of Fermat's Last Theorem comes to mind..
8. (Original post by HapaxOromenon3)
...
Is this not all just from MathOverflow? What is the point of typing it up here?
9. (Original post by ThatPerson)
Is this not all just from MathOverflow? What is the point of typing it up here?
It's interesting and many people won't have seen them before...
10. (Original post by HapaxOromenon3)
It's interesting and many people won't have seen them before...
Don't you think you should at least acknowledge your sources? Why not just post a link to that page instead?
11. This thread is embarrassing tbh. Ripping off other peoples comments with no credit and acting like you came up with them. Just delete this post please. What's next you gonna latex up FLT proof and claim it's yours.
12. (Original post by cliveb2016)
This thread is embarrassing tbh. Ripping off other peoples comments with no credit and acting like you came up with them. Just delete this post please. What's next you gonna latex up FLT proof and claim it's yours.
But what if he is Andy Andy Wilez . Unless you mean Fermats Little theorem, I came up with that 1.

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