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PRIME NUMBERS History, theories and applications By Kim Wojtowicz.

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1 PRIME NUMBERS History, theories and applications By Kim Wojtowicz

2 Definition of a Prime Number A Prime number is a number that has exactly 2 Distinct factors: itself and 1. Smallest prime number is 2, it is also the only even prime number.

3 Is 1 a Prime number? There is argument and controversy about 1 being Prime. Professor Ian Mallett states: DOES GOD THINK 1 Is Prime Does God think that the number 1 is a prime number? A good question, which can be answered with just four basic responses. 1. Yes! 2. No! 1 is a separate and special entity called ‘Unity’. It is neither a prime nor a composite number. The first prime number is 2. 3a. It doesn’t matter, it’s just a question of definition. 3b. It doesn’t matter, I don’t believe in God. 4. What is a prime number? In whatever category your reply falls, I’m sure you will benefit from reading this article so welcome to the site!

4 Dr. Mallet’s argument: Prime numbers are far more important than composite numbers in the worlds of mathematics, the sciences, cryptography etc. Mathematicians, mistakenly in my view, often refer to prime numbers as being the building bricks of all numbers. The act of building is an additive process rather than a multiplicative one. You can’t build a wall and then multiply that wall by four to make a house. You just have to keep building, adding one brick at a time, until the house is completed. The building brick of all numbers therefore is the number 1. The number 1 is the brick that builds the number n, where each value of n is equal to the number ‘bricks’ required. The number 10 requires 10 bricks or 10 1’s, 20 requires 20 1’s etc. Nonetheless, great importance is assigned to the prime number series. So why the confusion over their order numbers? Why isn’t there universal agreement as to which is the first prime? Surely this issue is at least equal to, and probably far more important than, the order number of the composite numbers

5 Up until the beginning of the last century the general consensus was that 1 is prime, with just a few detractors. The French mathematician Henri Lebesgue was emphatic on this issue. The majority of mathematical textbooks in the 19 th and early 20 th centuries gave 1 as prime. Even now, maths textbooks as late as 1995 or later show 1 as being the first prime. However, at the beginning of the last century mathematicians began to see the number 1 as a special case, isolating it from the primes and composites by calling it ‘unity’. It supposedly made things more ‘convenient’, an expression which I loathe because convenience doesn’t necessarily make a right and even now I still await an example of this so called ‘convenience’. The main opposition to the 1 is prime lobby is that it supposedly destroys that monolith of mathematics, the Fundamental Theorem of Arithmetic which states that every natural number is either prime or can be uniquely factored as a product of primes in a unique way - hmm, unique! Well, in the first instance isn’t the number itself unique? Can something be doubly unique? Of course not! Since the number itself is unique it would be logical that its prime factorisation is different to that of any other number. I don’t think it matters a lot whether or not this theorem is destroyed but there is no need for it to be destroyed if 1 is considered prime. It all boils down to definition again. By the insertion of just two words the theorem still stands: the Fundamental Theorem of Arithmetic states that every natural number is either prime or can be factored as a product of prime proper factors in a unique way.

6 Dr. Mallet’s Conclusion On a final note, how the idea that 1 is not prime destroys that beautiful Trinity of the first three primes 1, 2 and 3. The only triad of primes which are linked together, nothing in between, and the only primes whose order numbers are equal to their values. Uniquely, the product of these three is equal to their sum, i.e. 6, the 3 rd triangle. This is the gematria of the Greek word ABBA meaning Father or Daddy. The first three primes 1, 2 and 3 represent the Trinity in terms of the order of the Father, Son and Holy Spirit. Be not deceived dear reader, 1 is prime and God thinks so too! Personal Note: from what we have learned, math and religion don’t mix.

7 History of Primes Ishango bone: tool handle discovered around 1960 in the African area of Ishango, near Lake Edward. It has been dated to about 9,000 BC and was at first thought to have been a tally stick. At one end of the bone is a piece of quartz for writing, and the bone has a series of notches carved in groups on three rows running the length of the bone. The markings on two of these rows each add to 60. The first row is consistent with a number system based on 10, since the notches are grouped as , , , and , while the second row contains the prime numbers between 10 and 20! The Ishango bone is kept at the Royal Institute for Natural Sciences of Belgium in Brussels.prime numbers

8 Who had knowledge of Primes? Shown on Ishango bone Hints that Egyptians knew from the Rhind Papyrus. Ancient chinese writings show some knowledge Not much done with it till the Greeks circa 300BC.

9 Who has researched it? Eratoshenes nd-prime-sequential-sieves- algorithm/ nd-prime-sequential-sieves- algorithm/ mathematics_gifts_tshirts_or ganic_mens_tshirt_, mathematics_gifts_tshirts_or ganic_mens_tshirt_, eratosthenes_prime_guy_mu g, eratosthenes_prime_guy_mu g, /watch?v=0JHEqBLG650& NR=1&feature=fvwp /watch?v=0JHEqBLG650& NR=1&feature=fvwp /watch?v=3mrVyVBTYNc &feature=related /watch?v=3mrVyVBTYNc &feature=related

10 Competition for Erosthenes Arthur Oliver Lonsdale Atkin (July 31, 1925 – December 28, 2008), who published under the name A. O. L. Atkin, was a Professor Emeritus of mathematics at the University of Illinois at Chicago. As an undergraduate during World War II, he worked at Bletchley Park cracking German codes. [1] He received his Ph.D. in 1952 from the University of Cambridge, where he was one of John Littlewood's research students. [2]Professor Emeritus mathematicsUniversity of Illinois at ChicagoundergraduateWorld War II Bletchley ParkcrackingGerman [1]Ph.D.University of CambridgeJohn Littlewood [2] Atkin, along with Noam Elkies, extended Schoof's algorithm to create the Schoof–Elkies–Atkin algorithm and, together with Daniel J. Bernstein, developed the sieve of Atkin.Noam ElkiesSchoof's algorithmSchoof–Elkies–Atkin algorithmDaniel J. Bernstein sieve of Atkin

11 Sieve of Atkin In the algorithm: All remainders are modulo-sixty remainders (divide the number by sixty and return the remainder).modulo-sixty remaindersremainder All numbers, including x and y, are whole numbers (positive integers). Flipping an entry in the sieve list means to change the marking (prime or nonprime) to the opposite marking. Create a results list, filled with 2, 3, and 5. Create a sieve list with an entry for each positive whole number; all entries of this list should initially be marked nonprime. For each entry number n in the sieve list, with modulo-sixty remainder r : – If r is 1, 13, 17, 29, 37, 41, 49, or 53, flip the entry for each possible solution to 4x 2 + y 2 = n. – If r is 7, 19, 31, or 43, flip the entry for each possible solution to 3x 2 + y 2 = n. – If r is 11, 23, 47, or 59, flip the entry for each possible solution to 3x 2 − y 2 = n when x > y. – If r is something else, ignore it completely. Start with the lowest number in the sieve list. Take the next number in the sieve list still marked prime. Include the number in the results list. Square the number and mark all multiples of that square as nonprime. Repeat steps five through eight. This results in numbers with an odd number of solutions to the corresponding equation being prime, and an even number being nonprime.

12 Study/Theorems of Primes: Just to name a few Fermat: infinitely many primes. Gauss:The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Euclid:fundamental Theorem of Arithmetic Riemann: distribution of primes Goldbach: Every even number >2 can be expressed as the sum of 2 primes.

13 Types of Primes Twin Primes: primes differ by 2 Mersinne Primes: descibe Mn = 2 ^ ( n) -1 Fermat Primes: Fn = 2^ (2n) + 1 Prime Quadruplet: A prime constellation of four successive primes with minimal distance. The term was coined by Paul Stäckel ( prime constellationprimes Wieferich primes: prime p such that p squared divides 2^ ( p-1) -1 Wall-Sun-Sun Primes: if p squared divides F(p-p/5) Wilson Prime: p squared divides ( p-1)!

14 Prime Calculators est.php est.php b/primenumber/primecheck.htm b/primenumber/primecheck.htm Many others found on the internet

15 Largest known Prime In 2008 the largest known prime to date was found. Prime of record size:

16 Uses/applications of Primes htm Used for cryptography /insects htm estrial.pdf finding extraterrestrials estrial.pdf What is the best use of primes? AAgAYxy AAgAYxy

17 Music of the Primes Olivier Messiaen, a famous composer found a great use for prime numbers in his music: Messiaen used both a 17 and 29 sequence in his piece of music Quartet for the End of Time. Both motifs start at the same time, however, since they are both prime numbers, the same sequence of notes playing together from each sequence wont be the same until they have played through 17 x 29 times each. He held prime numbers very close to his heart and believed they gave his music a timelessness quality.

18 Primes In Nature Similarly, the cicada, a burrowing insect owes its survival to prime numbers and their properties. The cicada lives underground for 17 years, making no sound or showing any signs for this amount of time. After 17 years, all of the insects appear in the forest for just six weeks to mate before dying out.

19 Uses: Code Breaking Prime numbers assist us in internet banking, shopping and general interaction on the internet. It encodes the messages. Ex: RSA calculator: HW/RSAWorksheet.html HW/RSAWorksheet.html

20 Prime Numbers In Code Breaking To encode a message… If we want to send the message “HELLO” we simply convert it into a string of numbers: (A=01, B=02… etc.) We can then raise that number to a publicly announced power, divide it by another number which has again been publicly announced and we will be left with a remainder. This is our encoded message…

21 Prime Numbers In Code Breaking To decode this message… The person who received the coded string of numbers would raise that number to another power which would only be known to them. They then divide it again by the number publicly announced earlier and the remainder from that would be the string of numbers that break down to say “HELLO”!

22 Prime Numbers In Code Breaking For Example… Let our message be “E”. “E” is converted to 05, and is then raised to the 7 th power. Our number is now 78,125. We divide that number by 33 to give 2367 with a remainder of is our encoded message.

23 Prime Numbers In Code Breaking Now, to decode… We raise 14 to the 3 rd power to give We divide that number by 33 which gives 83 with a remainder of 5… 5 is our decoded message and converts to “E”, the original message.

24 Fun time! Now if time, a little fun with Relatively primes?

25 Prime Number excercise You take all people from a group, or all your class students and put them in a circle. Then count the number of students and pick a number “RELATIVELY PRIME to it” meaning that both numbers don’t have to be prime, they just need to share NO Factors

26 Game continued Example: lets say we have 20 students, and you pick the number 3. You give a Tennis ball to student 1 and they throw the ball around to every 3 rd person. It will Take 3 rotations, but in these 3 rotations the ball will come to each student 3 times, touching them all just once and then returning to where it started. You make it tougher by adding a new tennis ball to the circle everytime the ball passes to the 3 rd person start another ball going. 20 people should be able to handle 9 tennis balls (( # people/2 ) – 1)

27 On paper. You can do the same designs on paper as the game. Using a ruler, compass and carefully spacing out the dots connect them and create spirals and other designs. You will see as the two numbers get closer the angles shrink and the picture narrow. For our example to 20 you could use 20/3 and 20/7 and 20/9.

28 On paper On paper it gives wonderful designs for 2 number not relatively prime. For example pick 20/2 you will get 2 decagons that intertwine. 20/5 will give you 5 intertwining squares while 20/4 will give 4 intertwining decagons. Formula ( points / n where points and n are not relatively prime) gives n intertwining shapes of sides (point/n).

29 Resources 1)Does god think 1 is prime: _1_is_prime.htm _1_is_prime.htm 2)Internet encyclopedia of science: ber.html ber.html 3)RSA encryption calculator: orksheet.html orksheet.html 4)Wikipedia: 5)Prime curious:


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