The notion of proof lies at the very heart of maths. It's when it comes to proving things that mathematicians let loose their genius and creativity, and in the process often discover unexpected surprises or deep philosophical issues. But proofs can also be daunting. So to help you along, we've brought together a range of *Plus* articles on proofs, divided up into four groups:

- Proofs: what are they and why do we need them?: These articles explore the notion and role of proofs through the ages, with many interesting examples.
- Some nice proofs: These articles contain explicit example of proofs we particularly like because they are particularly elegant, surprising, or fun.
- Elusive proofs: In this category we look at proofs that have eluded mathematicians for centuries, exploring some famous unsolved (or recently solved) problems, and ways of attacking them.
- Philosophical proofs: It may seem as if maths is all about certainty, but there are actually many philosophical questions surrounding what constitutes a proof. What's more, there's proof that not everything can be proved. The articles in this category explore these limitations and questions.

If you'd like to get your hands dirty making your own proofs, then see the feature on proofs on our sister site NRICH. It contains problems and tasks that aim to introduce students to the formality and logic of mathematical proof.

### Proofs: what are they and why do we need them?

Why we want proof — What are mathematical proofs, why do we need them and what can they say about sheep?

The origins of proof — Part I — This article explores deductive reasoning and looks at the earliest known example of a proof.

The origins of proof — Part II — This article explains how the notion of proof was brought from mathematics into physics by the mathematician and astronomer Johannes Kepler.

The origins of proof — Part III — For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. This article explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems.

The origins of proof — Part IV — This article explores what a proof really *is*, and how we know that we've actually found one. One for the philosophers to ponder...

### Nice proofs

Seeing Pythagoras — The great thing about geometry is that you can sometimes prove things using pictures. Enjoy these three great visual proofs of Pythagoras' theorem!

Some lovely proofs by picture — Don't like trigonometry? Don't worry, here are three beautiful proofs of a well-known result that make do without it.

Maths in a minute: How many primes? — We've known since the time of the ancient Greeks that there are infinitely many prime numbers. Here's Euclid's elegant proof.

Maths in a minute: The square root of 2 is irrational — Here's one of the most elegant proofs in all of maths, and a great example of how proof by contradiction works.

Maths in a minute: Shake to solve — Looking at the same thing in two different ways can help you prove that two things are equal. Here's an example.

Trisecting the angle with origami — You can't do it with a compass and ruler, but you can do it with a few folds.

The art gallery problem — Sometimes a piece of mathematics can be so neat and elegant, it makes you want to shout "eureka!" even if you haven't produced it yourself. The proof of the at gallery problem is an example.

Friends and strangers — Can we always find order in systems that are disordered? An ingenious proof shows that often we can.

### Elusive proofs

**The Riemann hypothesis**

The Riemann hypothesis is one of the most famous problems in mathematics, whose proof has eluded mathematicians for 150 years. Here is are some articles exploring it in detail:

**Fermat's last theorem**

Fermat's last theorem — Fermat's last theorem had foxed mathematicians for over 400 years, when Andrew Wiles famously announced a proof in 1993. This collection of articles explores the theorem, its proof, and some related topics.

**The Poincaré conjecture**

The 100-year-old Poincaré conjecture made the headlines in 2006, unusually for a pure maths result. The reason was not so much the conjecture itself, but the strange character of the person who finally cracked it — Grigory Perelman even refused the substantial prize associated to proving the conjecture. The following articles explore the conjecture and track some of its exciting history:

- Mathematical millionaire?
- Exotic spheres, or why 4-dimensional space is a crazy place
- The Fields Medals 2006

**Harmless looking results about primes**

The primes are notorious for posing easy-looking questions that turn out to be fiendishly difficult to answer. The following articles look at some of these:

- Mathematical mysteries: The Goldbach conjecture
- Mathematical mysteries: Twin primes
- Mind the gap
- Elusive twins
- Primes without 7s
- From 70 million to just 600
- Twins move within reach

**Million dollar problems**

Some of the problems mentioned above are included in the set of the Clay Institute's *Millennium Prize Problems*: anyone solving them will earn one million dollars. The following articles explore this list of problems, which rank among the world's most difficult and important, and look at a further two examples.

- How maths can make you rich and famous
- How maths can make you rich and famous: Part II
- Code-breakers, doughnuts, and violins.
- Maths in a minute: The Navier-Stokes equations
- The travelling salesman

### Philosophical proofs

**Human versus machine**

Can a computer be trusted to prove a theorem? It's a question that divides the mathematical world. The following articles probe the arguments, and look at famous results that have been proved by computers, including the four colour theorem and Kepler's conjecture:

- The future of proof
- Will machine learning replace mathematicians?
- Welcome to the maths lab
- Country road, take me home
- Automated maths

**Infinite possibilities**

What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? The following articles explore how these questions gave rise to controversial maths which brought triumph to one man and ruin to another:

- A glimpse of Cantor's paradise
- Cantor and Cohen: Infinite investigators part I
- Cantor and Cohen: Infinite investigators part II

**Constructive maths**

Can maths exist if you can't see it? — Unlike physicists, mathematicians don't need to "see" something to be certain it exists. But are non-constructive proofs really valid? And what happens if you try and make do without them? This collection of articles explores a school of thought called constructivist mathematics.

**The limits of mathematics**

One of the most ground-breaking results in mathematics is known as *Gödel's incompleteness theorem*, which, loosely-speaking, states that there are limits to what can be proved mathematically. The following articles look at the theorem, its history and its legacy: