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2) Number Theory Factoids

3) Primitive Recursiveness of Computing Prime Numbers

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- 1. Theory of Computation History of Numbers, Number Theory Factoids, Primitive Recursiveness of Computing Prime Numbers Vladimir Kulyukin www.vkedco.blogspot.com
- 2. Outline ● ● History of Numbers: Are Arabic Numerals Really Arabic? Number Theory Factoids – – Prime Divisibility – Euclid Numbers – Euclid's 2nd Theorem – ● Well-Ordering Principle Euclid's Gift to Computer Science Computing Primes is Primitive Recursive
- 3. History of Numbers Are Arabic Numbers Really Arabic?
- 4. Origin of Decimals “The world owes the decimal notation to the Hindus, and arithmetic as a practical science would have been impossible without the decimal notation. The Arabs first learned that notation from the Hindus and introduced it into Europe.” Romesh Chunder Dutt. “A History of Civilization in Ancient India Based on Sanskrit Literature – Rationalistic Age” (1,000 BC – 242 BC)”
- 5. Nagari Script Numerals and Their Modern Equivalents 1 2 3 4 5 6 7 8 9 0
- 6. Sulva Sutra (8-th Century B.C.) Sulva Sutra is a Sanskrit document that dates back to 8th century B.C. ● How to find the value of a diagonal in relation to the side of the square? ● Sulva Sutra states: “Increase the measure by its third part, and this third by its own forth, less the thirty-fourth part of that fourth.” ● In other words, the square root of 2 is being computed as: ● 1 1 1 1+ + − = 1.4142156 3 3 ⋅ 4 3 ⋅ 4 ⋅ 34
- 7. Sulva Sutra (8-th Century B.C.) Modern approximation : 2 = 1.4142135 1 1 1 Sulva Sutra approximation : 1 + + − = 1.4142156 3 3 ⋅ 4 3 ⋅ 4 ⋅ 34 Keep in mind that the second approximation comes from an 8 - th century B.C. document.
- 8. Number Theory Factoids
- 9. Two Number Theory Concepts ● ● ● First and foremost, we will need two concepts from number theory before we discuss computation of primes and proving its primitive recursiveness Well-Ordering Principle Prime Divisibility: Every number has a prime divisor
- 10. Well-Ordering Principle ● ● ● The well-ordering principle states that every non-empty set of natural numbers has a smallest element In axiomatic set theory, the set of natural numbers is defined as the set that contains 0 and is closed under the successor operation The set of natural numbers {n | {0, …, n} is well-ordered} contains all natural numbers
- 11. Prime Divisibility For any natural number A > 1, there exists a prime number p such that p|A
- 12. Proof 1. Assume that the statement is not true 2. Then there exists a non-empty set K of natural numbers each of which is greater than 1 and has no prime divisors 3. By the Well-Ordering Principle, K has a smallest element s 4. s cannot be a prime, because it would have itself as a prime divisor 5. Thus, s is a composite number such that s = ab, where 1 < a < s and 1 < b < s 6. But, then a and b must have prime divisors 7. Thus, s has the same prime divisors as a and b 8. But this contradicts 2 and hence 1
- 13. Another Number Theory Factoid Let a, n, b be natural numbers. Then if a = n + 1, b > 1, b | a, then ¬( b | n ).
- 14. Euclid Numbers: Definition 2,3,5,7,11,13,17,... Let pi be the i - th prime. Ei = 2 ⋅ 3 ⋅ 5 ⋅ ... ⋅ pi + 1. Ei is the i - th Euclid number.
- 15. Euclid’s 2nd Theorem (Proposition IX.20 in Euclid’s Elements) There are infinitely many primes
- 16. Euclid’s 2nd Theorem Reformulated Given a finite sequence of primes 2,3,5,..., pi , Ei = 2 ⋅ 3 ⋅ 5 ⋅ ... ⋅ pi + 1 is either a new prime or has a prime divisor greater than pi .
- 17. Proof 1. Let there be a finite number of primes. 2. Let these primes be 2,3,5, ..., p i . 3. Consider the number E = 2 ⋅ 3 ⋅ 5 ⋅ ... ⋅ pi + 1. 4. E i is not divisible by any primes 2 ,3,5,...,pi . 5. E i is either a prime or a composite number. 6. If E i is a prime, we are done, because we found a new prime. 7. If E i is a composite, E i must have a prime divisor. 8. But from 4, it follows that the prime divisor must be greater than pi . 9. Thus, either E i is a prime greater than all the previous primes or it has a prime divisor which is greater than each of the previous primes.
- 18. Euclid’s 2nd Theorem: Another Formulation We set p 0 = 0 for the base case. Then we define pi to be the i - th prime. In other words, p1 = 2, p 2 = 3, p3 = 5, p 4 = 7,... Consider E = p n !+1. Then E is either a prime or is divisible by a prime greater than p n .
- 19. Proof If E is a prime, we are done. Assume that E is not a prime. Then E is composite. But then it must have a prime divisor greater than p n .
- 20. Lower & Upper Bounds for Computing Next Prime pi +1 ∈ [ pi + 1, pi !+1], because the next prime after pi is greater than pi . Hence, pi + 1 for the lower bound. But it must divide pi !+1. Hence, pi !+1 for the upper bound.
- 21. Examples p0 = 0 p1 = 2 ∈ [ p 0 + 1, p 0 !+1] = [1,2] p 2 = 3 ∈ [ p1 + 1, p1!+1] = [ 3,3] p 3 = 5 ∈ [ p 2 + 1, p 2 !+1] = [ 4,7] p 4 = 7 ∈ [ p3 + 1, p3 !+1] = [ 6,121]
- 22. Euclid's Gift to Computer Science ● ● ● ● What does Euclid's Theorem give computer scientists? It gives us the strict lower and upper bounds where we are guaranteed to find the next prime The existence of the strict upper bound immediately implies the applicability of bound minimalization Since bounded minimalization applies, it immediately implies the primitive recursiveness of computing primes
- 23. Computing Prime Numbers
- 24. Computing N-th Prime is Primitive Recursive Show that p n is primitive recursive
- 25. Recurrences 1. p 0 = 0 2. p n +1 = min [ Prime( t ) & t > p n ] t ≤ p n !+1
- 26. Formalizing Recurrences h(lower , upper ) = min [ Prime( t ) & t > lower ] k ( x ) = h( x, x!+1) t ≤ upper
- 27. Formalizing Recurrences 1. p 0 = 0 2. p n +1 = k ( p n )
- 28. References & Reading Suggestions ● ● Ch. 3, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal Romesh Chunder Dutt. A History of Civilization in Ancient India Based on Sanskrit Literature – Rationalistic Age (1,000 BC – 242 BC)

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