Some problems in Number Theory

Number theory - the study of the natural numbers - does not typically feature in school curricula, but it is a rich source of interesting problems which can lead to surprising results. The problems in this feature will offer you the opportunity to notice patterns, conjecture, generalise and prove results.

Many results can be explained by applying GCSE algebraic techniques, and many can also be explained using diagrams or pictorial representations.

Number theory is studied in more detail at university, and it is often seen as one of the "purest" forms of mathematics. Gauss said "Mathematics is the queen of the sciences - and number theory is the queen of mathematics".

You can watch a recording of the webinar in which we discussed the mathematical thinking which can be prompted by these problems.

problem

Always perfect

Age

14 to 18

Challenge level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

problem

Common divisor

Age

14 to 18

Challenge level

Can you find out what numbers divide these expressions? Can you prove that they are always divisors?

problem

Polite numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?