I would like to know about theorems which can give different proofs using completely different techniques.
Motivation:
When I read from the book Proof from the Book, I saw there were many many proof for the same theorem using completely different fields of mathematics.
(There were Six proofs of the infinity of primes based on different ideas even using topology.)
I wonder How mathematicians give this kind of proofs.
If you know such theorems and proofs please share them with me. Thank you.
13 Answers
$\begingroup$One example of a theorem with multiple proofs is the Fundamental Theorem of Algebra (All polynomials in $\mathbb{C}[x]$ have the "right number" of roots). One way to prove this is build up enough complex analysis to prove that every bounded entire function is constant. Another way is to build up algebraic topology and use facts about maps from balls and circles into the punctured plane. Both of these techniques are used specifically to show one such root exists (and once this is proved the rest of the proof is easy).
I think there are other possible proofs of the theorem but these are two I have seen.
$\endgroup$ 4 $\begingroup$The Brouwer fixed-point theorem. The Wikipedia lists the following methods:
- Homology
- Stokes's theorem
- Combinatorial (Sperner's lemma)
- Reducing to the smooth case and using Sard's theorem
- Reducing to the smooth case and using the COV theorem
- Lefschetz fixed-point theorem
- Using Hex
A large rectangle is tiled with smaller rectangles. Each of the smaller rectangles has at least one integer side. Must the large rectangle have at least one integer side?
What if you replace 'integer' with 'rational' or 'algebraic'?
Here are fourteen proofs.
$\endgroup$ 1 $\begingroup$As a kind of extreme example, I've heard of this book, reputed to have given over 360 proofs of the Pythagorean theorem. I'm not sure how pairwise different they are, but there's bound to be a lot of variety.
Another one that sticks in my mind is the proof of the Cayley-Hamilton theorem for real (or complex) matrices. There is an algebraic proof that works for any field at all, but for the reals and complexes, you can argue topologically that the matrix you are working with is the limit of a sequence of diagonalizable matrices, and since the approximations satisfy their characteristic polynomial, so does the limit.
$\endgroup$ 2 $\begingroup$$20$ different proofs 0f the Euler Formula$$\color{Red}{V-E+F=2.}$$
$10$ different proofs 0f the Gaussian Integral$$\color{Green}{\int_{\mathbb{R}}e^{-x^2}\,\mathrm{d}x=\sqrt\pi.}$$
A famous example is the law of quadratic reciprocity. Wikipedia says that several hundred proofs of the law of quadratic reciprocity have been found. Many details concerning proofs by different methods can be found at the question at MO.
$\endgroup$ $\begingroup$The first example that comes to my mind is the Cauchy-Schwarz inequality.
Here there is a paper I found some time ago in which the authors claim (I write claim, because I did not check each proof) to be able to show twelve different proofs of the result.
Actually, there is an entire book on inequalities that starts from this very basic one, with various proofs of it.
$\endgroup$ 0 $\begingroup$There are several results of this kind: For example the period three implies chaos theorem () was proved by Li and York (in 1975) by using the intermediate values theorem for continuous functions in despite the result belongs to discrete dynamics. A generalization of this result () was made before (in 1965) by A.N. Sharkovsky by using a very complicated and different method.
Also, I note that the most know theorems of L. Riemann were proved by introducing new ideas based on complex analysis. Those of Euler are based on the series...etc.
$\endgroup$ $\begingroup$Robin Chapman gives 14 proofs of $$\sum_{n=1}^{\infty}{1\over n^2}={\pi^2\over6}$$ here.
$\endgroup$ 1 $\begingroup$Irving Adler in 'a new look at geometry' gives a circular chain of proofs of alternate versions of the fifth postulate (that if a line striking each of two lines at the same angle, then the two lines do not cross at any distance).
The idea of such a proof shows that all of these propositions are identical in function.
$\endgroup$ $\begingroup$Proofs that $\sqrt{2}$ is irrational.
I know (at least) four, and there are lists of many more. I could go on, but would be a bore, so I'll let someone else have the floor.
$\endgroup$ 5 $\begingroup$For $n\geq k\in\mathbb{N}$, you can prove that $\dfrac{n!}{k!\cdot(n-k)!}$ is an integer from a combinatorial/counting argument. Establish that the formula gives the number of ways to choose a subset of $k$ from a set of $n$, and that automatically makes it an integer.
Or you can prove that $\dfrac{n!}{k!\cdot(n-k)!}$ is an integer by showing that the highest power of any prime $p$ dividing the denominator is less than the highest power of $p$ dividing the numerator.
Several proofs here.
$\endgroup$ 1 $\begingroup$Other examples of such many-proof theorems include:
1) Harmonic series diverges
There are at least 8 proofs of this fact based on different ideas. The first four of them can be found under this link:
The other four are of the following form: "If a subseries diverges, then the whole series diverges too. Now let's prove that the subseries of harmonic series, that formed only by reciprocals of primes diverges..."
And there are at least four different ways to prove it:
2) Variance of a random variable with support in $[0; 1]$ does not exceed $\frac{1}{4}$
There are at least three proofs of this, based on different ideas: What is the largest possible variance of a random variable on $[0; 1]$?
3) No group is both a direct product and a direct product of non-trivial groups
There are at least three proofs of this, based on different ideas: Does there exist a group that is both a free product and a direct product of nontrivial groups?
4) $(\mathbb{Q}, +)$ is not a direct product of any two non-trivial groups
There are at least three proofs of this, based on different ideas: Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?
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