Number Theory

What is Number Theory?

Number Theory is the study of relations between numbers. Until the 19th century, it was part of Arithmetic. Nowadays it is considered as a separate discipline by itself. It is also considered one of the difficult areas in math.

We can narrow the definition and say that Number Theory is the study of the whole numbers, with a particular interest in prime numbers.

The origin of some of the ideas in Theory of Numbers was the visualisation of properties & relations of numbers (covered in the previous two chapters) in terms of geometrical shapes like triangles & squares.

Importance of Number Theory

Number theory has many interesting results which can be understood even by primary school students.

It has a lot of opportunities for developing understanding and procedural fluency through many explorations. It can develop curiosity about numbers and math itself.

Let us see some elementary results of number theory which can be understood even by students in Primary School.

Pythagoras Triples - These are sets containing 3 numbers which exhibit a relation which can be described as a2 + b2 = c2

In any right angled triangle, the sum of the squares of the 2 shorter sides would always be equal to the square of the hypotenuse. When these 3 sides are expressible with integers, they are called Pythagorean Triples.

Some examples are 3, 4, 5 where 32 + 42 = 52 (9 + 16 = 25) and 5, 12, 13 where 52 + 122 = 132 (25 + 144 = 169).

Pythagoras Theorem in geometry states that a triangle which has the above measures for its 3 sides, would be a right angled triangle.

Irrationality of √2 -Euclid gave an elegant proof that √2 is an irrational number. We will deal with this when we come to the chapter 17.10 on irrational numbers.

Infinity of Primes - Euclid also gave a proof that there is no "greatest" prime number and that there are an infinity of them.

Fermat's Last Theorem - This theorem states that there is no extension of Pythagoras theorem for powers greater than 2.

Which means we cannot find a set of 3 numbers A, B & C such that An + Bn = Cn for any value of n greater than 2. This theorem and its proof are so interesting that they will be dealt with in chapter 32.4.

Ramanujan (Taxi Cab) Number - It states that 1729 is the smallest number which can be written as the sum of 2 cubes in 2 different ways. Note that it is slightly different from Fermat's Last Theorem.

1729 can be written as 13 + 123 OR as 93 + 103

The story behind this number is also an interesting incident in the history of math.

Goldbach's Conjecture

Any even number, greater than 2, can be written as a sum of 2 prime numbers. This conjecture has still not been proved.

Catalan's Conjecture

8 and 9 are the only two perfect powers of whole numbers, which are also consecutive. (since 8 = 23 and 9 = 32 ). This conjecture was proposed in 1844 and proved in 2002.

Properties of Numbers

We studied some simple properties of numbers in other chapters. Number Theorists try to identify many properties which are not easily apparent.

In 1770 Lagrange proved that every natural number (that is, 0, 1, 2, 3, 4, 5, 6, 7, …) can be written as a sum of four or fewer squares. For example, 14 = 9 + 4 + 1 which is 14 = 3²+2²+1²

Gauss’s Eureka Theorem. He showed that every natural number can be written as the sum of three triangular numbers.

Encyclopedia of Number Sequences/ Online Encyclopedia of Integer Sequences

Number theorists kept on discovering a number of interesting number sequences.

In 1973 mathematician Neil Sloane published his first encyclopedia, A Handbook of Integer Sequences, which contained about 2,400 sequences that also proved useful in making certain calculations.

In 1995 Neil Slone, together with his colleague Simon Plouffe, published The Encyclopedia of Integer Sequences , which contained some 5,500 sequences.

As of March 2023, the Online Encyclopedia of Integer Sequence (OEIS) contains a little more than 360,000 entries.

The Most Boring Number

As of March 2023, 20,067 was the smallest number that did not appear in any of the OEIS's stored number sequences. Hence it can be called the Most Boring Number!

Recreational Math

Section 34 of this book is on Recreational Math. Many results from Number Theory can be given to students as explorations in math.

They also provide the necessary practice for operational fluency in a pleasurable way. Hence they can be called "Happy Drills". Please see Chapter 66 for more details.