**Question**
Jamal notices that the difference between two consecutive square numbers is always an odd number.

Does this always work

Can you prove this conjecture

**Answer**

A useful starting point is to look at some consecutive square numbers as follows:

**Squares 1 4 9 16 25 36 49**

Differences 3 5 7 9 11 13
Tidy up presentation by showing arrows between each pair of square numbers as follows

The conjecture would seem to be true

To prove the conjecture, let the two consecutive square numbers be

*x*^{2} and (

*x*+1)

^{2}
You need to prove that the difference between x

^{2} and (

*x*+1)

^{2} is an odd number

The difference can be written as

(

*x*+1)

^{2} −

*x*^{2}
= (

*x*^{2} + 2

*x* + 1) −

*x*^{2}
= 2

*x* +1

Since the number 2

*x* is always an even number then 2

*x* + 1 must be an odd number so that the difference between two consecutive square numbers is always an odd number.

NOTE The conjecture can also be shown to be true diagrammatically as follows

There are many well known conjectures which mathematicians have tried to prove or disprove over the years.

Two such conjectures are:

**Goldbach's Conjecture** which says that "Every even integer greater than two can be expressed as the sum of two primes." For example 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 or 5 + 5 ...

The Twin Prime Conjecture which suggests that there are an infinite number of twin primes where a twin prime is a pair of prime numbers that have a difference of two. For example 3 and 5 , 5 and 7, 11 and 13, 17 and 19…. are all twin primes.

A list of other conjectures can be found

here.

A conjecture is a statement which would appear to be true, but has not yet been formally proven. When a conjecture is proved beyond doubt then it is called a theorem.

Until recently, the most famous conjecture was the (misnamed) Fermat's Last Theorem, which stated that if an integer n is greater than 2, then the equation an + bn = cn has no solutions (providing a, b, and c are not zero). .

The conjecture was finally proven in 1994 by Andrew Wiles, an English research mathematician. See

here for further information.