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Concept

Conjecture

A conjecture is an unproven statement based on observations of a pattern. It is an educated guess that holds true for many supporting cases. A conjecture about the sum of any three consecutive numbers, for example, is shown in the table.

Sum of Any Three Consecutive Integers
Observation I	Observation II	Observation III
$2 + 3 + 4 = 9 = 3 \cdot 3$	$7 + 8 + 9 = 24 = 3 \cdot 8$	$13 + 14 + 15 = 42 = 3 \cdot 14$
Conjecture: The sum of any three consecutive integers is three times the second number.

However, it is unknown whether any given conjecture holds true for all cases. It could be false under some circumstances and, therefore, cannot be used to support other claims. A counterexample is enough to prove that a conjecture is false. When a conjecture is rigorously proved, it becomes a theorem.

Extra

Goldbach's Conjecture

One of the best-known conjectures is Goldbach's Conjecture, named after the German mathematician Christian Goldbach.

Goldbach's Conjecture
Every even whole number greater than $2$ is the sum of two prime numbers.

For example, the even numbers below follow the rule.

142440 = 3 + 11 = 11 + 13 = 17 + 23

As of

2013,

the conjecture has been verified by a computer for all integers less than

4 \times 1 0^{18} .

In March

2000,

it was announced that anyone who could prove Goldbach's Conjecture and whose proof was accepted by other mathematicians would be awarded a one million dollar prize. Although the prize was kept open for two years, nobody claimed it.

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