Number Theory

1. Introduction

Numbers can be classified as Real and Complex. Further real numbers can be classified as integers, natural numbers, whole numbers, rational numbers, irrational numbers, prime numbers, composite numbers, co–prime numbers etc. Here is a brief summary of real numbers:

Set of all positive counting numbers or positive integers is known as set of natural numbers. Note that 0 is not considered a natural number.

For example {1, 2, 3, 4, 5,…..infinite}

Set of all positive counting numbers including zero is known as set of whole numbers. 

For example {0, 1, 2, 3, 4, 5,…..infinite}

Set of all numbers which can be positive, negative or zero is known as set of integers. 

For example {0, ±1, ±2, ±3, ±4,  ±5,…..infinite}. Integers which are multiple of 2 are said to be even integers otherwise odd.

If product of some integers is odd, all the integers are necessarily odd. If product of some integers is even, at least one integer is even. 

A prime number (or a prime) is a natural number which has exactly two distinct natural number divisors:  1 and number itself. Prime numbers are infinite. The first twenty–five prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Two is the only even prime number, the term odd prime refers to any prime number greater than two. There are 25 primes from 1 to 100 and there are 10 primes from 100 to 150. There are 11 primes from 151 to 200 

Prime numbers of the form \(2^n - 1\) are known as Mersenne primes. In order for \({2^n} - 1\) to be prime, it is necessary but not sufficient that n should be prime. For example \({2^3} - 1\) is prime but  \({2^{11}} - 1\) is not prime. \({2^{11}} – 1 = 2047\) and it is multiple of 23. Mersenne Primes are named after the seventeenth–century French monk Marin Mersenne. Fermat, a 17th century mathematician believed that the numbers \({2^n} + 1\) were always prime, if \(n\) is a power of 2. He had verified this for \(n = 1, 2, 4, 8\) and 16 and he knew that if \(n\) were not a power of 2, the result failed. Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case \({2^{32}} + 1 = 4294967297\) is divisible by 641 and hence \({2^{32}} + 1\) is not a prime number.

The sequence 1, 1, 2, 3, 5, 8, 13,…. is known as Fibonacci series. If nth term of the Fibonacci sequence is \({T_n}\), then \({T_n} = {T_{n - 1}} + {T_{n - 2}}\). Members of this series are known as  Fibonacci Numbers.  

Example 2: If \({T_n}\) denotes nth term of  a Fibonacci sequence, where 

\({T_n} = {T_{n - 1}} + {T_{n - 2}}\), where \(n \ge 3\)

If \({T_6}^2 - {T_5}^2 = 517\), find the value of \({T_8}\)

Solution: \({T_6}^2 - {T_5}^2 = 517\) 

or \(({T_6} + {T_5})({T_6} - {T_5}) = 47 \times 11\)

Hence \({T_6} = 29,\;{T_5} = 18\) \( \Rightarrow \) \({T_7} = 47,\;{T_8} = 76.\)

The integers \(a\) and \(b\) are said to be co–prime or relatively prime if they have no common factor other than 1 or, equivalently, if their greatest common divisor is 1. For example, 6 and 35 are co–prime, but 6 and 18 are not co–prime because they are both divisible by 6. The number 1 is co–prime to every integer.
Amicable numbers are two different numbers so related that the sum of the proper divisors of the one is equal to the other, one being considered as a proper divisor but not the number itself. Such a pair is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220.