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Number Theory A.D.M College for Women(A), Nagapattinam PG and Research Department of Mathematics Semester IV Core Course VIII Number Theory Class : II B.Sc (Mathematics) Prepared by : Mrs. M. Prabavathy, Assistant Professor in Mathematics. Miss.P.Udhaya Assistant Professor in Mathematics. COURSE OBJECTIVES: To Introduce the concepts of divisibility, prime number and prime-factorization. To learn about Euler Function, Greatest integer function and Mobius function. To know the conjectures in number theory. To acquire the knowledge of linear congruences. 5. To study the methods of classifying numbers. UNIT - I: Prime and Composite Numbers Coprimes-Sieve of Eratothenes-Euclid’s Theorem- Unique factorization-Fundamental Theorem of Arithmetic–Positional Representation of Integers-Number of Divisors-Sum of Divisors-Symbols 풅(풏), 흈(풏) -Arithmetic functions.(Chapter IV : Sec 77 to 97) ( Content hrs-9 hrs;Assignment-3 hrs) (12 Hours) UNIT - II: Prime and Composite Numbers Perfect Numbers- Euclid’s Theorem on Perfect Numbers-Amicable Numbers-Euler’s Function Φ(풏) - Greatest integer function-Mobious function (n) -Inversion formula and its converse.(Chapter IV: Sec 98 to 128) ( Content hrs-9 hrs;Assignment-3 hrs) (12 Hours) UNIT - III: Distribution of Primes General Discussion – Fermat’s Conjecture-Fermat Numbers-Gold Bach’ S Conjecture- Mersenne Numbers - GapTheorem-Infinitude of Primes.(Chapter V) ( Content hrs-9 hrs;Assignment-3 hrs) (12 Hours) UNIT - IV: Congruences Definition – Residue Classes - Complete and Least Residue Systems-Reduced Residue Systems – Casting out 9 – Magic Numbers- Divisibility Tests - Linear Congruences - Solution of Congruences - Chinese Remainder Theorem. (Chapter VI) ( Content hrs-9 hrs;Assignment-3 hrs) (12 Hours) UNIT - V: Quadratic Reciprocity Quadratic Residues and Non Residues-Euler Criterion-Primitive Roots is a Quadratic Non Residue-Legendre symbol-Gauss lemma-Quadratic Reciprocity Law. (Chapter X: Sec 255 to 278) ( Content hrs-9 hrs;Assignment-3 hrs) (12 Hours) TEXT BOOK : Prof. S.Kumaravelu and Susheela Kumaravelu, Elements of Number Theory , Raja Sankar off set Printers ,Sivakasi,2002. REFERENCE BOOKS: 1. David M. Burton, Elementary Number Theory , W.M.C. Brown Publishers, Dubuque, Lawa, 1989. 2. George E. Andrews, Number Theory, Hindhustan Publishing Corporation, 1984. COURSE OUTCOMES: On the completion of the course, learners will be able to • find the divisor, sum and product of a given natural number. • gain the knowledge of number theoretic functions. • interpret the Famous conjectures in number theory . • Solve the System of linear congruences using the Chinese Remainder theorem. • Apply the Law of Quadratic Reciprocity to classify numbers as quadratic residues and quadratic non-residues 1 Unit I Definition 1 A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. But,the factors of 29 are 1 and 29. That is, only two factors. Hence 29 is a prime number, but 28 is not a prime number. Note Prime numbers are numbers that have only 2 factors: 1 and themselves. The following are prime numbers: 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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. Definition 2 A composite number is a positive integer that has more than two positive integer factors,other than 1 and itself. For example, if we list the factors of 36, we have 1, 2, 3, 4,6,9, 12,18, and 36. That is, nine factors. Definition 3 Positive integers having no common factors are together called coprimes. They are also said to be prime to each other or relatively prime. (e.g.) 3 and 10 are coprimes since they have no common factors other than 1. Note: 1) A Co-prime number is a set of numbers or integers which have only 1 as their common factor i.e. their highest common factor (HCF) will be 1. 2) Co-prime numbers are also known as relatively prime or mutually prime numbers. It is important that there should be two numbers in order to form co-primes. 3) Co-primes are not necessarily prime numbers. Any two primes are coprimes. If 푎 and 푏 are coprimes, then (푎, 푏) = 1. Definition 4 If 푝 is a prime and 푝 + 2 is also a prime, then they are called twin primes. Examples: 3,5 and 11,13 are twin primes. Definition 5 If two adjacent integers are prime, then they are called Siamese twins. The integers 2 ad 3 are the only Siamese twins. The integers 2 and 3 are the only Siamese twins. The Sieve of Eratosthenes Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you use to drain spaghetti when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind. To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the first one hundred positive integers (1-100): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 Unit I 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Cross out 1, because it is not prime. Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number. Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2. Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number. Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100. The Sieve method is given in the following steps: Write down all the positive integers in their natural order upto the desired number. Cross out 1, because it is not prime. Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number. Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2. 3 Unit I Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number. Continue this process of striking out the multiples of the early prime numbers. Theorem The number of primes is infinite. Proof: Suppose there are only a finite number of primes 푝1, 푝2, 푝3, … , 푝푛 . Take 푃 = 1+푝1푝2푝3 … 푝푛 . Clearly, 푃 is not divisible by any of the primes 푝1, 푝2, 푝3, … , 푝푛 . Therefore, either 푃 is itself a prime or it is divisible by a prime other than the above said 푛 primes. This implies that the number of primes is not finite. Theorem The set of prime numbers is countable. Proof: The prime numbers taken in order are 1,2,3,5,7,11,13,17,19,23,… These can be considered as the first,second, third, etc primes and may be denoted by 푡ℎ 푝1, 푝2, 푝3, … , 푝푛. Thus the 푛 prime is 푝푛 and so on. So the primes can be arranged as a sequence {푝푛} and hence the set of primes is countable. Unique factorization theorem Any positive integer > 1 can be uniquely expressed as the product of prime numbers. Proof: Let 푛 > 1 be any positive integer. If 푛 is itself a prime number then 푛 stands as a product with a single prime factor. If 푛 is not a prime number it can be factorized so that we get 푛 = 푎1푎2 where 푎1 < 푛 & 푎2 < 푛. If 푎1 is a prime let it stand. If 푎1 is not a prime, it can be factorised as 푏1푏2 where where 푏1 < 푛 & 푏2 < 푛. Similarly for 푎2. The process of writing each composite number that arises as a product of factors must terminate at some stage because the factors are smaller than the given composite number and each factor is an integer greater than 1. Thus we 푠 find 푛 can be written as a product of prime numbers in the form 푛 = 푝1푝2 … 푝푟 where all the 푝푖 are not necessarily different. Let us prove that the factorization is unique. If possible let 푛 be factorized as prime factors in two different ways so that 푛 = 푝1푝2 … 푝푟 and 푛 = 푞1푞2 … 푞푠 where all the 푝푟 are different from 푞푠. We get 푞1푞2 … 푞푠 푝1푝2 … 푝푟 = 푞1푞2 … 푞푠 ⇒ 푝2 … 푝푟 = ∈ ℤ ⇒ 푝1|푞1푞2 … 푞푠 푝1 which is impossible unless 푝1is one of 푞1푞2 … 푞푠. Similarly every 푝푖is some 푞푠 ⇒ the two factorization of 푛 are identical. Hence the factorization is unique. Note: 푠 (i) Let 푛 = 푝1푝2 … 푝푟 be the prime factorization of 푛.
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