VEDIC MATHEMATICS : Primes

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VEDIC MATHEMATICS Primes

• T. K. Prasad
• http//www.cs.wright.edu/tkprasad

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Divisibility

• A number n is divisible by f if there exists
  another number q such that n f q.
• f is called the factor and q is called the
  quotient.
• 25 is divisible by 5
• 6 is divisible by 1, 2, and 3.
• 28 is divisible by 1, 2, 4, 7, 14, and 28.
• 729 is divisible by 3, 9, and 243.

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Prime Numbers and Composite Numbers

• A prime number is a number that has exactly two
  factors 1 and itself.
• Smallest prime number is 2.
• 1 is not a prime number.
• Examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
  etc.
• A composite number is a number that has a factor
  other than 1 and itself.
• 1 is not a composite number.

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First 100 primes

• 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
  191 193 197 199 211 223 227 229 233 239 241 251
  257 263 269 271 277 281 283 293 307 311 313 317
  331 337 347 349 353 359 367 373 379 383 389 397
  401 409 419 421 431 433 439 443 449 457 461 463
  467 479 487 491 499 503 509 521 523 541

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Eratosthenes and the Primes

• Eratosthenes of Cyrene (276 B.C. - 194 B.C.,
  Greece) was a Greek mathematician, poet, athlete,
  geographer and astronomer.
• Eratosthenes was the librarian at Alexandria,
  Egypt.
• He made several discoveries and inventions
  including a system of latitude and longitude. He
  was the first person to calculate the
  circumference of the Earth, and the tilt of the
  earth's axis.
• Eratosthenes devised a 'sieve' to discover prime
  numbers.

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Sieve
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The Sieve of Eratosthenes

• Algorithm to enumerate primes n
• Generate the sequence 2 to n
• Print the smallest number in the remaining
  sequence, which is the new prime p.
• Remove all the multiples of p.
• Repeat 3 and 4 until the sequence is exhausted.

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Hundreds Chart
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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1 Cross out 1 it is not prime.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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2 Leave 2 cross out multiples of 2
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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3 Leave 3 cross out multiples of 3
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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4 Leave 5 cross out multiples of 5
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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5 Leave 7 cross out multiples of 7
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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6Leave 11 cross out multiples of 11
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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All the numbers left are prime
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
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
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The Prime Numbers from 1 to 100 are as follows
2,3,5,7,11,13,17,19, 23,29,31,37,41,43,47, 53,59,6
1,67,71,73, 79,83,89,97
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Perfect Number

• A perfect number is a number which is equal to
  the sum of its (proper) factors.
• Examples 6, 28, 496, 8128, etc
• 1 2 3 6
• 1 2 4 7 14 28
• These were the only perfect numbers known to
  early Greek mathematicians (500 BC).

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Amicable Numbers

• Amicable numbers are pairs of numbers such that
  the sum of the proper factors of one is equal to
  the other.
• Example (220, 284)
• Proper factors of 220 are 1, 2, 4, 5, 10, 11, 20,
  22, 44, 55 and 110, which sum to 284 and
• Proper factors of 284 are 1, 2, 4, 71, and 142,
  which sum to 220.
• Amicable and perfect numbers were known to the
  Pythagoreans (500 BC).

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Prime Decomposition

• Every natural number greater than one has a
  unique prime factorization. That is, it can be
  uniquely expressed as a product of prime numbers.
  
• E.g.,
• 120 2 2 2 3 5
• 981189 3 3 11 11 17 53
• 3141879 3 13 13 6197

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Proof that Primes are infinite Proof by
Euclid (300 B.C. )

• Let us assume that the set of primes is finite.
  Primes 2, 3, , p
• Consider the number n (2 3 p) 1.
• Claim n is a prime but is not in Primes.
• Reason Each prime divides the first summand but
  not 1, so it will not divide n. Hence, n is a new
  prime not in Primes!
• Conclusion Primes are not finite.

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Advanced Material

• FYI

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Perfect Numbers

• Euclid (300 BC) discovered a general formula for
  even perfect numbers.
• 2(n - 1) (2n - 1) is a perfect
  number
• whenever (2n - 1) is a (Mersenne)
  prime.
• Verify that for n 2, 3, 5, and 7,
• you get 6, 28, 496, and 8128,
  respectively.
• Fifth perfect number is 33550336, for n 13.
• (211 - 1) is not a prime because 2047 23 89.

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Demonstrating perfection!

• Prove 2(n - 1) (2n - 1) is a perfect number,
• whenever (2n - 1) is a prime.
• Proof Sum of factors
  
• 2(n - 1) 2(n - 2) 2 1
• (2n - 1) 2(n - 2)
  2 1
• 2n - 1
• (2n - 1) 2(n - 1) - 1
  (see next slide)

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Auxiliary Result

• Show
• 2(n - 1) 2(n - 2) 2 1 2n -
  1
• Let S 2(n - 1) 2(n - 2) 2 1
• 2 S 2n 2(n - 1) 22 2
• 2 S - S 2n 1
• S 2n - 1

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(contd)

• Proof Sum of factors
  
• 2n - 1 (2n - 1) 2(n - 1) -
  1
• (2n - 1) 1 2(n - 1) - 1
• (2n - 1) 2(n - 1)
• (original number)

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Open problems in Number Theory

• Goldbach's conjecture Every even integer greater
  than 2 can be written as the sum of two primes.
• Odd perfect numbers It is unknown whether there
  are any odd perfect numbers.
• Observe Factoring large primes is a very hard
  problem so a number of cryptographic systems are
  based on that fact.

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Primes Generation in Scheme

• (define (interval-list m n)
• (if (gt m n) '()
• (cons m (interval-list ( 1 m) n))))
• (define (primeslt n)
• (sieve (interval-list 2 n)))
• (primeslt 300)

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(contd)

• (define (sieve l)
• (define (remove-multiples n l)
• (if (null? l) '()
• (if ( (modulo (car l) n) 0) division
  test
• (remove-multiples n (cdr l))
• (cons (car l)
• (remove-multiples n (cdr l))))))
• (if (null? l) '()
• (cons (car l)
• (sieve (remove-multiples (car l) (cdr l))))))

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Perfection in Python

• def perfectNumber(n)
• (factorList, factorSum) (,0)
• for i in range(1, 1 (n / 2)) help(math)
• if ( (n i) 0 )
• factorList.append(i)
• factorSum i
• if n factorSum
• return (n, factorList)
• else
• return False