Complicated proofs of simple facts...

Theorem: The sum of the degrees of the vertices of a graph is even.
Proof: The number

N

of graphs with degrees

d_1,\ldots,d_n

is the coefficient of

x_1^{d_1}\cdots x_n^{d_n}

in the generating function

\prod_{j<k}(1+x_jx_k)

. Now apply Cauchy's Theorem in

n

complex dimensions, giving

N = \frac{1}{(2\pi i)^n} \oint\cdots\oint \frac{\prod_{j<k}(1+x_jx_k)}{x_1^{d_1+1}\cdots x_n^{d_n+1}} dx_1\cdots dx_n

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

x_j=e^{i\theta_j}

, we get

N = \frac{1}{(2\pi)^n} \int_{-\pi}^\pi\cdots\int_{-\pi}^\pi \frac{\prod_{j\lt k}(1+e^{\theta_j+\theta_k})}{e^{i(d_1\theta_1+\cdots +d_n\theta_n)}}d\theta_1\cdots d\theta_n

Alternatively, choosing the circles

x_j=e^{i(\theta_j+\pi)}

, we get

N = \frac{1}{(2\pi)^n} \int_{-\pi}^\pi\cdots\int_{-\pi}^\pi \frac{\prod_{j\lt k}(1+e^{\theta_j+\theta_k})}{e^{i(d_1\theta_1+\cdots +d_n\theta_n+k\pi)}}d\theta_1 \cdots d\theta_n

where

k=d_1+\cdots+d_n

. Since

e^{ik\pi}=-1

when

k

is an odd integer, we can add these two integrals to get

2N=0

.

Theorem:

5!/2

is even.
Proof:

5!/2

is the order of the group

A_5

(see http://groupprops.subwiki.org/wiki/Alternating_group:A5). It is known that

A_5

is a non-abelian simple group. Therefore

A_5

is not solvable. But the Feit-Thompson Theorem asserts that every finite group with odd cardinal is solvable, so

5!/2

must be an even number.

Theorem:

2^{1/n}

is irrational for

n\geq3

.
Proof: If

2^{1/n}=p/q

then

p^n = q^n+q^n

, contradicting Fermat's Last Theorem. Unfortunately FLT isn't strong enough to prove the irrationality of

\sqrt{2}

though.
Alternative Proof:

X^n-2

is an irreducible polynomial, by Eisenstein's Criterion.

Theorem: The harmonic series diverges.
Proof: Assume it is convergent. Then the functions

f_n := \frac{1}{n} 1_{[0,n]}

would be dominated by an absolutely integrable function. But

\int_{\bf R} \lim_{n \to \infty} f_n(x)\ dx = 0 \neq 1 = \lim_{n \to \infty} \int_{\bf R} f_n(x)\ dx

contradicting the Dominated Convergence Theorem.

Theorem: There are infinitely many primes.
Proof:

\zeta(3)=\prod_p \frac{1}{1-p^{-3}}

is irrational, by Apery's Theorem.

Theorem: All integers

n>1

can be written as the sum of two squarefree integers.
Proof: We can manually check that the result holds for

n \le 10^4

. Now let

S

be the set of all squarefree integers, except the primes larger than

10^4

. Then using
(a) the fact that the Schnirelmann density of the set of squarefree integers is

\dfrac{53}{88}

- 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-counting_function#Inequalities
we have that the Schnirelmann density of

S

is greater than

\dfrac{1}{2}

. By Mann's Theorem, we now have that every positive integer can be written as the sum of at most

2

elements of

S

. In particular, every prime number, and every non-squarefree integer, can be written as the sum of exactly

2

elements of

S

. Now all that remains is to prove the theorem for composite squarefree integers, which can be done as follows:

n = pq = (p_1 + p_2)q = p_1q + p_2q

, where

p

is the smallest prime dividing

n

and

p_1, p_2

are squarefree integers.

Reply 1

username1162972

18

Nvm

(edited 7 years ago)

Reply 2

username1162972

18

Oh this wasn't a question sorry my bad. Ignore the above as you probably know it.

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Reply 3

Math12345

14

Euclid's proof of infinitely many primes is much more elegant...

Reply 4

cliveb2016

2

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?

Reply 5

username1162972

18

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.

Reply 6

math42

20

Proof of Fermat's Last Theorem comes to mind..

Reply 7

ThatPerson

18

Original post by HapaxOromenon3

...

Is this not all just from MathOverflow? What is the point of typing it up here?

Reply 8

HapaxOromenon3

OP

3

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...

Reply 9

Zacken

22

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?

Reply 10

cliveb2016

2

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.

Reply 11

username1162972

18

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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Complicated proofs of simple facts...

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