A square number (Fig.3.3) is an integer that is the square of another integer. For example, the number 4 is a square number since 4 = 22. Similarly, the number 9 and the number 16 are square numbers. A numbernis a square number if and only if one can arrange thenpoints in a square. For example, the square numbers 4, 9, 16 are represented in squares as follows:
The square of an odd number is odd, whereas the square of an even number is even. This is clear since an even number is of the formn = 2kfor somek, and so n2= 4k2which is even. Similarly, an odd number is of the formn = 2k+ 1 and so n2= 4k2+ 4k+ 1 which is odd.
A rectangular number (Fig.3.4)n may be represented by a vertical and hori- zontal rectangle ofn points. For example, the number 6 may be represented by a rectangle with length 3 and breadth 2, or a rectangle with length 2 and breadth 3.
Similarly, the number 12 can be represented by a 43 or a 34 rectangle.
A triangular number (Fig.3.5)nmay be represented by an equilateral triangle of npoints. It is the sum ofknatural numbers from 1 tok. = That is,
n¼1þ2þ þk
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+ =
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Fig. 3.2 Pythagorean triples
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Fig. 3.3 Square numbers
Parity of Integers
The parity of an integer refers to whether the integer is odd or even. An integernis odd if there is a remainder of one when it is divided by two, and it is of the form n= 2k+ 1. Otherwise, the number is even and of the formn= 2k.
The sum of two numbers is even if both are even or both are odd. The product of two numbers is even if at least one of the numbers is even. These properties are expressed as
eveneven¼even evenodd¼odd oddodd¼even eveneven¼even evenodd¼even oddodd¼odd
Divisors
Leta andbbe integers witha6¼0 thena is said to be a divisor ofb (denoted by a|b) if there exists an integerksuch thatb =ka.
A divisor ofnis called atrivial divisorif it is either 1 ornitself; otherwise it is called anontrivial divisor. Aproper divisorofnis a divisor ofnother thannitself.
Definition (Prime Number)
Aprime numberis a number whose only divisors are trivial. There are an infinite number of prime numbers.
Thefundamental theorem of arithmeticstates that every integer number can be factored as the product of prime numbers.
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Fig. 3.5 Triangular numbers
56 3 Number Theory
Mesenne Primes
Mersenne primes are prime numbers of the form 2p−1, wherep is a prime. They are named after Marin Mersenne (Fig.3.6) who was a 17th French monk, philosopher and mathematician. Mersenne did some early work in identifying primes of this format, and there are 47 known Mersenne primes. It remains an open question as to whether there are an infinite number of Mersenne primes.
Properties of Divisors (i) a|band a|cthen a|b+ c (ii) a|bthena|bc
(iii) a|band b|c thena|c
Proof (of i) Supposea|b anda|c thenb= k1aand c= k2a.
Then,b +c= k1a +k2a = (k1+k2)a and soa|b+ c.
Proof (of iii) Suppose a|b andb|c thenb= k1aand c=k2b.
Then,c=k2b = (k2k1)aand thus a|c.
Perfect and Amicable Numbers
Perfect and amicable numbers have been studied for millennia. A positive integer mis said to beperfectif it is the sum of its proper divisors. Two positive integers mand n are said to be an amicable pair if m is equal to the sum of the proper divisors ofn and vice versa.
Aperfect numberis a number whose divisors add up to the number itself. For example, the number 6 is perfect since it has divisors 1, 2, 3 and 1 + 2 + 3 = 6.
Perfect numbers are quite rare and Euclid showed that 2p−1(2p− 1) is an even perfect number whenever (2p− 1) is prime. Euler later showed that all even perfect numbers are of this form. It is an open question as to whether there are any odd perfect numbers, and if such an odd perfect numberNwas to exist thenN> 101500. A prime number of the form (2p− 1), where p is prime called as Mersenne prime. Mersenne primes are quite rare and each Mersenne prime generates an even perfect number and vice versa. That is, there is a one to one correspondence between the number of Mersenne primes and the number of even perfect numbers.
Fig. 3.6 Marin Mersenne
It remains an open question as to whether there are an infinite number of Mersenne primes and perfect numbers.
Anamicable pairof numbers is a pair of numbers such that each number is the sum of divisors of the other number. For example, the numbers 220 and 284 are an amicable pair since the divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, which have sum 284, and the divisors of 284 are 1, 2, 4, 71, 142, which have sum 220.
Theorem 3.1 (Division Algorithm)For any integer a and any positive integer b there exist unique integers q and r such that
a¼bqþr 0r\b
Proof Thefirst part of the proof is to show the existence of integers qandrsuch that the equality holds, and the second part of the proof is to prove uniqueness of qand r.
Consider…−3b,−2b,−b, 0,b, 2b, 3b,…then there must be an integerqsuch that
qba\ðqþ1Þb
Thena −qb= rwith 0 r<band soa= bq+rand the existence ofqandris proved.
The second part of the proof is to show the uniqueness ofqandr. Supposeq1 and r1also satisfya =bq1+r1with 0 r1<b and suppose r<r1. Thenbq+ r=bq1+r1and sob(q −q1) =r1−rand clearly 0 < (r1− r) <b.Therefore,b|
(r1− r) which is impossible unless r1−r= 0. Hence,r= r1andq =q1. Theorem 3.2 (Irrationality of Square Root of Two)The square root of two is an irrational number (i.e., it cannot be expressed as the quotient of two integer numbers).
Proof The Pythagoreans3 discovered this result and it led to a crisis in their community as number was considered to be the essence of everything in their
3Pythagoras of Samos (a Greek island in the Aegean sea) was an influential ancient mathematician and philosopher of the sixth century B.C. He gained his mathematical knowledge from his travels throughout the ancient world (especially in Egypt and Babylon). He became convinced that everything is number and he and his followers discovered the relationship between mathematics and the physical world as well as relationships between numbers and music. On his return to Samos he founded a school and he later moved to Croton in southern Italy to set up a school. This school and the Pythagorean brotherhood became a secret society with religious beliefs such as reincarnation and they were focused on the study of mathematics. They maintained secrecy of the mathematical results that they discovered. Pythagoras is remembered today for Pythagoras’s Theorem, which states that for a right-angled triangle that the square of the hypotenuse is equal to the sum of the square of the other two sides. The Pythagorean’s discovered the irrationality of the square root of two and as this result conflicted in a fundamental way with their philosophy that number is everything, and they suppressed the truth of this mathematical result.
58 3 Number Theory
world. The proof is indirect: i.e., the opposite of the desired result is assumed to be correct and it is showed that this assumption leads to a contradiction. Therefore, the assumption must be incorrect and so the result is proved.
Suppose ffiffiffi p2
is rational then it can be put in the form p/q, where p and q are integers andq 6¼0. Therefore, we can choosep,qto be co-prime (i.e., without any common factors) and so
p=q ð Þ2¼2 )p2=q2¼2 )p2¼2q2 )2jp2 )2jp )p¼2k )p2¼4k2 )4k2¼2q2 )2k2¼q2 )2jq2 )2jq
This is a contradiction as we have chosen p and q to be co-prime, and our assumption that there is a rational number that is the square root of two results in a contradiction. Therefore, this assumption must be false and we conclude that there is no rational number whose square is two.