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Named and Notorious Primes Joe Frost University of Washington Computing and Communications Joyce Frost Lake Washington High School.

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1 Named and Notorious Primes Joe Frost University of Washington Computing and Communications Joyce Frost Lake Washington High School

2 Prime Numbers “Prime numbers are the very atoms of arithmetic... The primes are the jewels studded throughout the vast expanse of the infinite universe of numbers that mathematicians have studied down the centuries.” Marcus du Sautoy, The Music of the Primes

3 A History and Exploration of Prime Numbers Dedicated to Royal Penewell -1923-2008 Math Teacher and Prime Enthusiast

4 Named and Notorious Primes Early Primes Named Primes Hunting for Primes Harnessing Primes

5 Euclid of Alexandria 325-265 B.C. The only man to summarize all the mathematical knowledge of his times. In Proposition 20 of Book IX of the Elements, Euclid proved that there are infinitely many prime numbers.

6 Eratosthenes of Cyrene 276-194 B.C. Librarian of the University of Alexandria. Invented an instrument for duplicating the cube, measured the circumference of the Earth, calculated the distance from the Earth to the Sun and the Moon, and created an algorithm for finding all possible primes, the Eratosthenes Sieve.

7 Nicomachus of Gerasa c. 100 A.D. Introduction to Arithmetic, Chapters XI, XII, and XIII divide odd numbers into three categories, “prime and incomposite”, “composite”, and “the number which is in itself secondary and composite, but relatively to another number is prime and incomposite.” In chapter XIII he describes Eratosthenes’ Sieve in excruciating detail.

8 Pierre de Fermat 1601-1665 Fermat’s Little Theorem - If a is any whole number and p is a prime that is not a factor of a, then p must be a factor of the number (a p-1 -1). Mentioned in a letter in 1640 with no proof, proved by Euler in 1736

9 Leonhard Euler 1707-1783 Euler proved a stronger version of Fermat’s Little Theorem to help test for Euler Probable Primes: “If p is prime and a is any whole number, then p divides evenly into a p -a.”

10 Carl Friedrich Gauss 1777-1855 At 15, he received a table of logarithms and one of primes for Christmas He noticed that primes are distributed to approximately π(N) ~ N/log(N), now called The Prime Number Theorem First mentioned it in a letter 50 years later.

11 Bernhard Riemann 1826-1866 One of the million-dollar problems is the Riemann Hypothesis: "All non-trivial zeros of the zeta function have real part of one half." ζ (s) = ∑ (n -s ) (n=1,2,3,…) or ζ (s) =∏(n s )/(n s -1) (n=2,3,5,7,11,…)

12 Named and Notorious Primes Early Primes Named Primes Hunting for Primes Harnessing Primes

13 Absolute Prime Also called permutable prime, an absolute prime is a prime with at least two distinct digits which remains prime on every rearrangement (permutation) of the digits. For example, 337 is a permutable because each of 337, 373 and 733 are prime. Most likely, in base ten the only permutable primes are 13, 17, 37, 79, 113, 199, 337, and their permutations.

14 Cullen Primes Fr. James Cullen, SJ, was interested in the numbers n*2 n +1 (denoted C n ). He noticed that the first, C 1 =3, was prime, but with the possible exception of the 53rd, the next 99 were all composite. Later, Cunningham discovered that 5591 divides C 53, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141.

15 Cullen Primes of the Second Kind Five decades later Raphael Robinson showed C 141 was a prime. The only known Cullen primes C n are those with n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, and 481899. These numbers are now called the Cullen numbers. Sometimes, the name "Cullen number" is extended to also include the Woodall numbers: W n =n*2 n -1. These are then the "Cullen primes of the second kind".

16 Fermat Primes Fermat numbers are numbers of the form Fermat believed every Fermat number is prime. F n is prime for F n is composite for 4 < n < 31, but no one knows if there are infinitely many Fermat Primes.

17 Euler PRP Euler was able to prove a stronger statement of Fermat’s Little Theorem which he then used as to test for Euler probable primes. If an Euler PRP n is composite, then we say n is an Euler pseudoprime.

18 Ferrier’s Prime Ferrier’s Prime is the largest prime found before electronic calculators. Ferrier’s Prime = 1/17(2 148 +1) = 20988936657440586486151264256610222 593863921

19 Fibonacci Prime A Fibonacci prime is a Fibonacci number that is prime. 1,1,2,3,5,8,13,21,34,55,89,144…

20 Sophie Germain Prime A Sophie Germain prime is a prime p such that q=2p+1 is also prime - (2, 3, 5, 11, 23, …) Around 1825, Sophie Germain proved that the first case of Fermat's last theorem is true for such primes, i.e., if p is a Sophie Germain prime, then there do not exist integers x, y, and z different from 0 and none a multiple of p such that x p +y p =z p.

21 Goldbach’s Conjecture “Every even number is a sum of two primes.” Has been verified for all even numbers to 400 trillion, but not yet proved.

22 Illegal Primes Phil Carmody published the first known illegal prime. When converted to hexadecimal, the number is a compressed form of the computer code to crack CSS scrambling. It is "illegal" because publishing this number could be considered trafficking in a circumvention device, in violation of the Digital Millenium Copyright Act.

23 Lucas Prime A Lucas prime is a Lucas number that is prime. The Lucas numbers can be defined as follows: v 1 = 1, v 2 = 3 and v n+1 = v n + v n-1 (n > 2) Lucas numbers are like Fibonacci numbers, except that they start with 1 and 3 instead of 1 and 1.

24 Mersenne Prime Mersenne primes are the primes of the form 2 n –1. Mersenne claimed that n in {2,3,5,7,13,19,31,67,127,257} would yield primes A Gaussian Mersenne prime is a prime using Gaussian integers (1, -1, i, -i).

25 Landry and Aurifeuille The mathematician Landry devoted a good part of his life to factoring 2 n +1 and finally found the factorization of 2 58 +1 in 1869 (so he was essentially the first to find the Gaussian Mersenne with n=29). Just ten years later, Aurifeuille found the Gaussian factorization, which would have made Landry's massive effort trivial.

26 Lucas-Lehmer Number The Lucas-Lehmer test is an efficient deterministic primality test for determining if a Mersenne number M_n is prime. A Mersenne Number 2 n -1 is prime if it divides the Lucas-Lehmer number L n where L n =(L n-1 ) 2 -2

27 Palindromic Prime A palindromic prime is a prime that is a palindrome. A pyramid of palindromic primes by G. L. Honaker, Jr. 2 30203 133020331 1713302033171 12171330203317121 151217133020331712151 1815121713302033171215181 16181512171330203317121518161 331618151217133020331712151816133 Largest known is 10 5901 1465410 5901 1

28 Royal Prime Royal Primes are primes where the digits are all prime and a prime can be constructed through addition or subtraction using all the digits. These are named after Royal Penewell, treasurer of the Puget Sound Council of Teachers of Mathematics (PSCTM) from 1973 to 2005 and who was born in `23, the first Royal Prime of the century.

29 Repunit Primes Repunits are positive integers in which all the digits are 1, denoted as R 1 = 1, R 2 =11, etc. Of these, the following are known to be prime:11, 1111111111111111111, and 11111111111111111111111 (2, 19, and 23 digits), R 317 (10317-1)/9, and R 1,031 (101031-1)/9. In 1999 Dubner discovered that R 49081 = (1049081-1)/9 was a probable prime, in 2000 Baxter discovered the next repunit probable prime is R 86453, and in 2007 Dubner identified R 109297 as a probable prime.

30 Twin Primes Twin Primes are primes whose difference is 2. Conjectured but not proven that there are an infinite number of twin primes. All twin primes except (3, 5) are of the form 6n+/-1. 2486!!!!+/-1 are twin primes with 2151 digits

31 Cousin Primes Cousin primes are primes whose difference is 4. The first few pairs are {3,7},{7,11},{17,23},{43,47}

32 Sexy Primes Sexy primes are primes whose difference is 6. The first few sexy primes pairs are {7,13}, {11,17}, {13,19}, and {17,23}

33 Wieferich Prime By Fermat's Little Theorem any prime p divides 2 p-1 -1. A prime p is a Wieferich prime if p 2 divides 2 p-1 -1. In 1909 Wieferich proved that if the first case of Fermat’s last theorem is false for the exponent p, then p satisfies this criterion. Since 1093 and 3511 are the only known such primes (and they have been checked to at least 32,000,000,000,000), this is a strong statement! In 1910 Mirimanoff proved the analogous theorem for 3 but there is little glory in being second. Such numbers are not called Mirimanoff primes.

34 Named and Notorious Primes Early Primes Named Primes Hunting for Primes Harnessing Primes

35 How Many Primes? Euclid proved there are infinitely many primes N=(AxBxCx…P)+1, N>A,B,C…P. If N prime, then it is larger than the others and not included in the list. If N is composite, then one of (A,B,C…) divides N, and divides N-(AxBxC…) which is 1, which is impossible. QED

36 Gauss and Legendre Gauss noticed the frequency of primes approached N/log(N) but didn’t publish. Legendre noticed that the frequency of primes approaches N/(log(N)-1.80366) and published in 1808, finding that yet again, Gauss had been there first. How Many Primes.xls

37 Prime Number Theorem Gauss mentioned in a letter, but did not prove, that the number of primes less than x can be approximated by: Proved independently by Jacques Hadamard of France and Charles de la Vallee Poussin of Belgium in 1896

38 Peter Gustav Lejeune-Direchlet Direchlet used Euler’s connection of primes to the zeta function to prove Fermat’s conjecture about infinitely many primes modulo 1 to any base Zeta function - values can be calculated as ζ (x) = 1/1 x +1/2 x +1/3 x +…1/n x +…

39 Density Function Gauss introduced π(x)= # of primes less than or equal to x Riemann showed that the zeta function can also be written as a product over its zeroes in the complex plane:

40 Riemann’s Hypothesis Fourier’s technique of adding waveforms to model complex graphs, Cauchy’s weird world of complex numbers, and Direchlet’s fascination with Euler’s zeta function are basic to Bernhard Riemann’s conjecture: “The real part of any non-trivial zero of the Riemann zeta function is 1⁄2.”

41 Prime Number Sieves Eratothsenes Sieve Excel Sieves Quadratic Sieve Number Field Sieve

42 Quadratic Sieve Data collection phase computes a congruence of squares modulo the number to be factored Data processing phase uses Gaussian elimination to reduce a matrix of the exponents of prime factors of the remainders found in the data collection phase.

43 Number Field Sieve An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers.

44 Great International Prime Search Great International Mersenne Prime Search lets anyone with a computer be part of the search for the next record-setting prime. In November, 2001, Canadian student Michael Cameron used his PC to prove the primality of 2 13,466,917 -1, the 39th Mersenne Prime. Five more have been discovered since then.

45 Opportunity On September 4, 2006, Dr. Curtis Cooper and Dr. Steven Boone's CMSU team discovered the 44th known Mersenne prime, 2 32,582,657 -1. Edson Smith using GIMPS found 2 43,112,609 -1 (about 12.9 million digits, Aug 08), winning the $100,000 prize from the Electronic Freedom Foundation

46 Prime Generators There are several polynomial functions that generate primes for a while before they start yielding composite numbers. F(x) = x 2 + x + 41 yields prime number for x < 40.

47 Generating all primes No polynomial known which generates all and only primes, but this generates only primes and negative numbers: F(a,b,…z) = (k + 2)(1 - (wz + h + j - q) 2 - ((gk + 2g + k + 1)(h + j) + h - z) 2 - (2n + p + q + z - e) 2 - (16(k + 1) 3 (k + 2)(n + 1) 2 + 1 - f 2 ) 2 - (e 3 (e + 2)(a + 1) 2 + 1 - o 2 ) 2 - ((a 2 - 1)y 2 + 1 - x 2 ) 2 - (16r 2 y 4 (a 2 - 1) + 1 - u 2 ) 2 - (((a + u 2 (u 2 - a)) 2 - 1)(n + 4dy) 2 + 1 - (x + cu) 2 ) 2 - (n + l + v - y) 2 - ((a 2 - 1)l 2 + 1 - m 2 ) 2 - (ai + k + 1 - l - i) 2 - (p + l(a - n - 1) + b(2an + 2a - n 2 - 2n - 2) - m) 2 - (q + y(a - p - 1) + s(2ap + 2a + p 2 - 2p - 2) - x) 2 - (z + pl(a - p) + t(2ap - p 2 - 1) - pm) 2 )

48 Elliptic Curve Factorization Faster than the Pollard rho factorization and Pollard p-1 factorization methods. (Wolfram website)

49 Named and Notorious Primes Early Primes Named Primes Hunting for Primes Harnessing Primes

50 Prime Factorization Every number can be expressed as a unique product of prime numbers. Example: 450 = 2*3*3*5*5

51 Greatest Common Factors The Greatest Common Factor is the product of the list of shared factors. Example: 450 = 125 = GCF(125,450) = 5*5 2*3*3*5*5 5*5*5

52 Least Common Multiple The Least Common Multiple can be found by writing the prime factorizations of both numbers and crossing off one copy of the set that forms the Greatest Common Factor.

53 Testing Processors In 1995, Nicely discovered a flaw in the Intel® PentiumTM microprocessor by computing the reciprocals of 824633702441 and 824633702443, which should have been accurate to 19 decimal places but were incorrect from the tenth decimal place on.

54 Communication In Carl Sagan’s novel Contact, aliens send a series of prime numbers to show intelligence behind radio transmissions

55 Quantum Physics The frequency of the zeroes of the Riemann zeta function appears to match the energy levels in the nucleus of a heavy atom when it is being bombarded with low-energy neutrons. Freeman Dyson noticed the similarity at a chance meeting with mathematician Hugh Montgomery.

56 Quantum Physics, II German Sierra and Paul Townsend will publish a paper in Physical Review Letters that suggests that an electron constrained to move in two dimensions and constrained by electric and magnetic fields have energy levels that match the zeros of the zeta function.

57 Winning Bets Don Zager, who argued against Riemann’s Hypothesis, bet two bottles of wine that an exception would be found in the first 300,000,000 roots. A Dutch team calculated an extra 100 million roots to help win the bet. Those were the most expensive bottles of wine, ever.

58 RSA Encryption Ron Rivest, Adi Shamir, and Len Adleman harnessed Fermat’s Little Theorem to enable secure web communications Fermat’s Little Theorem: if p is prime and a is an integer not divisible by p, then (a p-1) =1(mod p). Factoring large numbers is computationally difficult

59 Named and Notorious Primes Joyce Frost - Joe Frost -

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