Famous Conjectures

Conjecture, Proof & Counter Examples are 3 powerful ideas in math.

Math is the study of patterns formed by numbers, shapes & events. A Conjecture is a guess based on the study of such patterns. Once a conjecture is proposed, many people interested in math try to prove it.

It remains a conjecture until it is either proved right or proved wrong with a counterexample.

A valid logically correct explanation of a conjecture is called a proof. It is a series of statements tied together by rigorous math logic. Every step should be logically derivable from the previous step. To say that a conjecture holds good for a few numbers or even a million numbers is not good enough.

The idea of the Pythagoras Theorem which connects all three sides of a right-angled triangle through areas is one such example. The theorem has been proved to the satisfaction of all mathematicians.

If the process of trying to prove a conjecture yields even a single counterexample, then the conjecture is not valid anymore! It is called a counterexample because it's an example that counters, or goes against, the statement's conclusion.

There have been conjectures which were proved after many years, in some cases, centuries. The Fermat's Last Theorem was a conjecture which took almost 250 years for confirming it with a proof.

There are conjectures which have not been proved till today. The Collatz Conjecture & the Goldbach's conjecture are examples.

Some conjectures were also proved wrong. One of the earliest examples is the conjecture of the Pythagorean school that all numbers can be expressed as a ratio of 2 whole numbers. This conjecture was found to be wrong with the discovery of the square root of 2. This was a dramatic discovery which resulted in the discovery of irrational numbers.

Some of the unproved conjectures in math are world-famous. Many of them "about are also simple enough that the problem itself can be understood by students of primary school! Let us see some of them.

Collatz Conjecture

1. Write down a number (in initial stages write a small number to get practice)

2. Work out the next number as per the following rules

   1. If the number is even, then divide it by 2

   2. If the number is odd then multiply it by 3 and add one. In short find 3n+1

3. Write down the next number below the previous number

4. Repeat steps 2 & 3 until you come to a surprising end.

This number series was discovered in 1937 by Lother Collatz. But till today no one has been able to prove why the numbers behave as they do. It is considered the simplest unsolved problem in mathematics.

Four Color Theorem

All of us know about the political map of a country or a state which has been divided into smaller areas like states or districts. If we want each of the smaller divisions to be visible clearly, we colour each area with a different colour. No two divisions which share a common border should have the same colour. What is the minimum number of colours required?

In 1852, a college student in London, wondered if 4 was the number. Many people who tried with different kinds of maps, also came up with the answer 4. But till today there is no mathematical proof that the minimum number of colours for colouring any map is 4!

With the invention of computers, many tried to solve the problem with a computer. In 1976, a computer worked out that there were only 1936 different kinds of maps and each of them requires only 4 colours.

But math being what it is, the computer proof was not accepted and the world is still waiting for a mathematical proof!

Goldbach Conjecture

In 1742, Christian Goldbach conjectured that every even number greater than 2, can be written as a sum of 2 prime numbers.

The Conjecture gained substantial recognition after Goldbach shared it in a letter to the renowned Swiss mathematician Leonhard Euler. Although Euler responded positively and expressed his belief in the conjecture’s validity, rigorous proof eluded both of them.

With computers, mathematicians have been able to show that this conjecture is true for all numbers up to 400,000,000,000,000!

But as in the case of the Four-Color Theorem, the computer proof was not accepted, and the world is still waiting for a mathematical proof!

Twin Prime Conjecture

We know that all primes are odd numbers. Two consecutive odd numbers which are also primes are called Twin Primes. Some examples are 5 & 7, 17 & 19, 29 & 31 etc.

Euclid proved that the number of primes is infinite. Mathematicians have been trying to see if Twin Primes are also infinite. Though this is believed to be true, no one has been able to prove it.

Just for interest, the largest twin primes discovered till today are

2,003,663,613X2195,000 and 2,003,663,613X2195,000 + 1 which have 58711 digits!

Honeycomb Conjecture

It says that the regular hexagonal grid is the best way to divide a surface into regions of equal area with the least total perimeter.

The origin of this problem is obscure; it is mentioned in a text of Marcus Terentius Varro around 36 B.C.

The mathematical proof was provided in 1999 by T. Hales. So it has taken 2,035 years to prove it!

The theorem and generalizations thereof have immediate applications in optimizing space, physical structures, and material waste, for instance in construction.

Bees seem to use this conjecture instinctively!

With the hexagonal shape, the bees consume the least amount of wax for a given honeycomb.

This seems to be evidence of mathematical ideas & structures being embedded in the minds of all living beings by the process of evolution.