# What are the theorems in mathematics which can be proved using completely different ideas?

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.

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.

• See more proofs in the book The Fundamental Theorem of Algebra by Fine and Rosenberger. – lhf Jan 26 '15 at 10:33
• You can prove the fundamental theorem of algebra using Lie theory: if $L / \Bbb C$ were a finite extension, the complex projective space $PL$ would inherit the structure of a compact commutative Lie group from the multiplication on $L$. But compact commutative Lie groups are tori, and complex projective space in any positive dimension is not a torus. (Credit goes to Mariano Suárez-Alvarez; I learned this proof from a comment he posted somewhere.) You can also prove the FTA entirely algebraically, though I don't remember the details. – user98602 May 6 '15 at 3:29
• See mathoverflow.net/questions/10535/… where 44 answers have been posted. – Gerry Myerson Jul 15 '15 at 7:20
• There is an algebraic proof that makes use of the Sylow theorems and basic Galois theory. – Slade Jul 15 '15 at 7:52

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
• What is COV theorem? – Mizar Jan 28 '15 at 9:10
• @Mizar, change of variables: en.wikipedia.org/wiki/…. – Martín-Blas Pérez Pinilla Jan 28 '15 at 9:14
• While we're at it, there are two proofs using the Brouwer degree for continuous maps of open sets in $\mathbb{R}^n$. One is essentially the same as Hirsch's theorem based on the impossibility of a retraction onto $S^{n-1}$, and the other constructs a homotopy between the identity and identity minus the continuous map. – Gyu Eun Lee Mar 16 '15 at 7:14

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.

• Thanks, this is very nice! – joriki Jul 27 '15 at 17:29

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.

• +1, and interestingly, the geometric proof of Cayley-Hamilton even carries over to arbitrary fields when one uses the Zariski topology. – Hanno Jan 18 '15 at 13:15
• I'm currently this this, which gives a number of different proofs of the Pythagorean Theorem. Some proofs are very similar to each other, but others are quite different. – Randy E Jan 18 '15 at 13:16
1. $20$ different proofs 0f the Euler Formula $$\color{Red}{V-E+F=2.}$$

2. $10$ different proofs 0f the Gaussian Integral $$\color{Green}{\int_{\mathbb{R}}e^{-x^2}\,\mathrm{d}x=\sqrt\pi.}$$

• This and this also provides some methods of evaluating Gaussian integral. – Bumblebee Jul 15 '15 at 8:17

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.

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.

There are several results of this kind: For example the period three implies chaos theorem (http://mathworld.wolfram.com/PeriodThreeTheorem.html) 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 (http://mathworld.wolfram.com/SharkovskysTheorem.html) 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.

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.

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.

Robin Chapman gives 14 proofs of $$\sum_{n=1}^{\infty}{1\over n^2}={\pi^2\over6}$$ here.

• This article provides a beautiful non-proof (Euler's proof) of this identity. Thank you. – Bumblebee Jul 17 '15 at 8:50

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.

• You can also prove this using group actions: $S_n$ acts on the set $\{1, 2, \dotsc, n\}$, and the stabilizer of any $k$-set from that set is $S_k\times S_{n-k}$. So by the orbit-stabilizer theorem, the number of such $k$-sets is the number of orbits, which is $\frac{|S_n|}{|S_k||S_{n-k}|} = \frac{n!}{k!(n-k)!}$. – rogerl Jul 27 '15 at 17:27

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: https://proofwiki.org/wiki/Harmonic_Series_is_Divergent

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: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes

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?