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# All, None and Forever

July 7, 2017

There is something very magical about unsolved mathematical problems, especially those that have been around for hundreds of years. Teachers at school don't have a solution for them and to date, the world's greatest mathematicians don't have solutions either. Some of these problems are quite intricate and require a lot of background to comprehend. But others, are very simple and crisp in their statement, even though solutions are currently not in reach.

When a problem is easy to describe but very hard to solve there is something very appealing to it. It means that you can describe it to young curious minds and get them thinking about the current limits of human knowledge. When we present such problems to young souls, we often see yearning expressions and excitement. It isn't uncommon to see ambitious students wondering: “Will I ever do it?”. “Will it be done in my lifetime?”. "Can it be done?". Related questions may be: "Have civilizations on other planets found the solution?".  Or maybe, "Has someone on this planet found the solution but kept it secret?".

What makes a problem simple to state? Perhaps it is the fact that hardly any mathematical background is needed to understand the problem. For example, you may just need to know a thing or two about numbers. But before that, you need to understand the language used in a mathematical statement so that you can make sense of what the problem is actually asking. The language often involves these key words:

All, None and Forever.

What do we mean by this? Open problems often require you to prove or disprove one of the following types of statements:

All: A property holds in all cases.

None: There do not exist objects that satisfy a property.

Forever: A property appears in a repetitive pattern forever.

To get a feel for "All, None and Forever", let's visit three very famous problems that seem simple even though they are extremely difficult to solve. Specifically, we will explore the Goldbach Conjecture, Fermat's Last Theorem and the Twin Prime Conjecture. We certainly won't go into the depth of the historical development of the problems, but rather get a feel of what they say.

If you know a little bit about number theory -- the study of whole numbers, you may say, "Hey, I know Fermat's Last Theorem. It isn't open! It was actually solved and proved to be true!". Indeed you are right, a solution was provided by Andrew Wiles in the 1990's after years of work and research effort. So why are we discussing it here?

The Proof of Fermat's Last Theorem serves as an example of a problem that occupied attention of top mathematicians for centuries (ever since 1637).

It could only be resolved after an incredible amount of work that hundreds if not thousands of researchers, continuously and collectively developed.

The other two problems that we present are currently unresolved. Anyone who will solve solve them would be world famous. Will they be resolved in our lifetime? Maybe, but no one knows for sure.

Let's now see All, None and Forever in action through these three famous problems.

Goldbach Conjecture (All): Every positive even number, m, greater than 2 can be written as a sum of two prime numbers, p and q:

This states that if you have a positive even number, say 16, you can then write it as a sum of two prime numbers.

You may first ask: What is a prime number? A prime number is a whole number greater than 1 that is not divisible by any other number except for 1 and itself. There are infinitely many of them, 2, 3, 5, 7, 11, 13, 17, 19,... and this list goes on forever in a rather erratic manner.

You may then ask: How can I sum two prime numbers (p, q) and get 16 (m)? Try p=5 and q=11. There may even be other options. Do you see one?

It may seem easy if we choose one value for m, but the problem is that we need to find two primes, p and q, whose sum is m for all even numbers m not just one. You can understand more about Goldbach's conjecture from this Numberphile video:

Fermat's Last Theorem (None): There do not exist positive whole numbers, a, b and c, that satisfy:

where n is a positive whole number greater than 2.

You may associate Fermat's Last Theorem to something that you may be more familiar with --the Pythagoras' Theorem. Here is an example:

The numbers (3, 4, 5) are called a Pythagorean triple which satisfies the top equation when n = 2. Some of us even know this by heart. But when you go on to n = 3, strangely as it may seem, you won't find any (a,b,c) triple of whole numbers that satisfies the equation. See this Numberphile video for more:

Twin Prime Conjecture (Forever): There are infinitely many pairs of prime numbers that differ by 2 . In a mathematical equation, this can be represented as:

where p and q are primes. Such primes are called twin primes.

Examples include the twins 5 and 7, the twins 59 and 61 or the twins 137 and 139. The conjecture (or belief) is that no matter how far we go down the number line, we'll find more and more twin primes.

Computers sometimes give insight into such problems. While, generally, computers can't provide proofs, they are used for numerical exploration. As of September 2016, the currently largest known twins are 2,996,863,034,895 · 2^1,290,000 ± 1. These numbers have 388,342 digits if they were to be written out! This is an impressive number, but it doesn't really tell us if the twin primes go forever or not.

This video by James Tanton of the Global Math Project, first explores the infinity of the primes and then builds up to the enigma of the twin primes. Do they go on forever?

There is so much to discover when you start thinking about mathematics. Sometimes, simple mathematical statements require so much effort to prove. Mathematicians around the world are continuously working on these open problems with passion and a full heart, hoping that one day, the secret will be unraveled.

Revolutionary educators are also seeking ways to bring human intelligence together. The Global Math Project, provides a platform where curious minds discuss math and spark ideas across the world. This year, the theme is exploding dots, a simple yet beautiful way of doing arithmetic and algebra. You don't need a math degree to participate in the Global Math Week (10th - 17th Oct); you certainly don't have to prove anything. All you need is to revisit the way that you learned about arithmetic at school. Give the content of the Global Math Project a try. You'll be amazed.

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