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**Named and Notorious Primes**

Canadian Student Michael Cameron’s 39th Mersenne Prime Joe Frost University of Washington Computing and Communications Joyce Frost Lake Washington High School

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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 Marcus du Sautoy is a professor of Mathematics at University of Oxford and a research fellow at the Royal Society.

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**A History and Exploration of Prime Numbers**

Dedicated to Royal Penewell Math Teacher and Prime Enthusiast to One of the founders of PSCTM, WSMC, and NWMC Taught in the Bellevue School District for many years

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**Named and Notorious Primes**

Early Primes Named Primes Hunting for Primes Harnessing Primes

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**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. Below is a proof closer to that which Euclid wrote, but still using our modern concepts of numbers and proof. See David Joyce's pages for an English translation of Euclid's actual proof. Theorem. There are more primes than found in any finite list of primes. Proof. Call the primes in our finite list p1, p2, ..., pr. Let P be any common multiple of these primes plus one (for example, P = p1p2...pr+1). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. Notice p can not be any of p1, p2, ..., pr, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete.

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**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. Prime pages has birth in 271 BC Prime pages says sieve is more efficient than table of stored primes for small N

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**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. His book is one of the few surviving documentations of Greek number theory Also wrote “Manual of Harmonics”. This is the first important music theory treatise since the time of Aristoxenus and Euclid. It provides the earliest surviving record of the story of Pythagoras's epiphany outside a smithy that pitch is determined by numeric ratios. Nicomachus also gives the first in depth account of the relationship between music and the ordering of the universe via the "music of the spheres."

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Pierre de Fermat 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 (ap-1-1). Mentioned in a letter in 1640 with no proof, proved by Euler in 1736 Fermat was a lawyer and judge, who spent a lot of time in chambers to avoid plaintiffs. If p is prime and (a,p) = 1, then ap-1=1 (mod p). Euler’s proof is on pages , using the binomial theorem (from Newton) Number theory took a back seat to the new field of Calculus for much of the late 17th and early 18th century. William Dunham , Journey through Genius, p225

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Leonhard Euler 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 ap-a.” Journey through Genius - His collected works are 70 volumes. As a child, he studied on Saturdays with Johann Bernoulli. Was appointed to a position in medicine in St. Petersburg at 20, stayed 14 years, then came back 25 years later. His publication backlog was 47 years at his death and his works and articles were 1/3rd of all mathematics published from His works are still not all published. Euler apparently enjoyed having his 13 children around him, and even managed to carry out mathematical researches with a baby on his lap! (Dunham) The number e, the base of the natural logarithms is named after him. (Derbyshire, 2003)

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Carl Friedrich Gauss 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. Called the Prince of Mathematics, he is perhaps the greatest mathematician of all time, but he was Euler's opposite on publication. His diaries are full of discoveries, but he published relatively little. Famous stories include learning to read and correcting his father's accounts at age 3, adding numbers from 1 to 100 in minutes, noticed by Duke of Brunswick at age 14 who paid for his schooling. Had to choose between philology and math at age 18, and was influenced by Fermat's work to choose math.

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Bernhard Riemann 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) =∏(ns)/(ns -1) (n=2,3,5,7,11,…) Shy, hypochondriac and sickly, he switched from theology to mathematics while at Gottingen at age 20, then transferred to Berlin University, where he finished his doctorate by age 25. However shy he was in person, he was extraordinarily bold in mathematics. Gauss was still teaching, but as little as possible (Derbyshire, 2003)

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**Named and Notorious Primes**

Early Primes Named Primes Hunting for Primes Harnessing Primes

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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.

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Cullen Primes Fr. James Cullen, SJ, was interested in the numbers n*2n +1 (denoted Cn). He noticed that the first, C1=3, was prime, but with the possible exception of the 53rd, the next 99 were all composite. Later, Cunningham discovered that 5591 divides C53, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141. Father James Cullen, S.J. (April 19, 1867 – December 7, 1933) was born at Drogheda, County Louth, Ireland. He studied mathematics at the Trinity College, Dublin for a while, but eventually turned to theology and was ordained on July 1, 1901. In 1905, he taught mathematics at Mount St. Mary's College in Derbyshire and published his finding of what is now known as Cullen numbers in number theory.

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**Cullen Primes of the Second Kind**

Five decades later Raphael Robinson showed C141 was a prime. The only known Cullen primes Cn are those with n=1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, , , and These numbers are now called the Cullen numbers. Sometimes, the name "Cullen number" is extended to also include the Woodall numbers: Wn=n*2n -1. These are then the "Cullen primes of the second kind". Raphael M Robinson, husband of Julia Bowers Robinson, first woman president of the American Mathematical Society. Coded the Lucas-Lehmer primality test on a SWAC, a vacuum tube computer with bit words of memory built in In 1952, Robinson used the SWAC to discover five Mersenne primes—the largest prime numbers known at the time, with 157, 183, 386, 664, and 687 digits

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**Fermat Primes Fermat numbers are numbers of the form**

Fermat believed every Fermat number is prime. Fn is prime for Fn is composite for 4 < n < 31, but no one knows if there are infinitely many Fermat Primes. F0=3, F2=17, F3=257, f4= 65537, f5=4,294,967,297 = x*y (determining x and y are an exercise for the student)

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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. In number theory, a probable prime (PRP) is an integer that satisfies a specific condition also satisfied by all prime numbers. The test is usually chosen to make exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a coprime to n and calculate a^(n − 1) modulo n. If the result is different from 1, n is composite. If it is 1, n is a probable prime Euler's test is: .

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Ferrier’s Prime Ferrier’s Prime is the largest prime found before electronic calculators. Ferrier’s Prime = 1/17(2148+1) =

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**Fibonacci Prime A Fibonacci prime is a Fibonacci number that is prime.**

1,1,2,3,5,8,13,21,34,55,89,144…

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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 xp+yp=zp. Sophie Germaine, Studied math in her father’s library. When her parents forbade her to explore mathematics, she smuggled books into her room and read by candlelight. When caught, her parents took away the candles, and for good measure, her clothes as well, expecting that she wouldn’t be able to study in a cold, dark room. Unable to attend courses, she eavesdropped outside classrooms and borrowed lecture notes from sympathetic students. She corresponded with Gauss about his Disquisitiones Arithmaticae. She was voted an honorary degree from Gottingen in 1831, but died before it was awarded. - Journey through Genius

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**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. Christian Goldbach was an amateur mathematician “whose enthusiasm exceeded his ability.” His chief claim to fame is his correspondence with Euler, including his famous 1742 conjecture. Goldbach's correspondence with Euler sparked Euler's interest in Fermat’s assertions and Euler produced four volumes of proofs of Fermat’s theorems. The proof of Goldbach's Conjecture, however, is still not found, although Estermann proved in 1929 that it is almost always true and Schnirelman proved in 1931 that any even number can be written as the sum of not more than 300,000 primes. (Dunham, 1990)

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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.

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Lucas Prime A Lucas prime is a Lucas number that is prime. The Lucas numbers can be defined as follows: v1 = 1, v2 = 3 and v n+1 = vn + v n-1 (n > 2) Lucas numbers are like Fibonacci numbers, except that they start with 1 and 3 instead of 1 and 1. Edouard Lucas ( ) set a record for the largest prime in The record stood for 75 years. See also Lucas-Lehmer numbers. Also invented the Towers of Hanoi puzzle. Died of infection of a cut from a piece of a plate dropped at the annual dinner of the French Association for the Advancement of Sciences. Dick Lehmer ( ) is credited with introducing computing as an experimental science. He was fired from Berkeley for refusing to sign the loyalty oath in the McCarthy era, but rehired when the oaths were declared unconstitutional.

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Mersenne Prime Mersenne primes are the primes of the form 2n–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). Father Marin Mersenne ( ) corresponded with many mathematicians of his time, including his schoolmate Rene Descartes. He acted as the clearinghouse for discovery in his day. Notice that 2^257 is 77 digits and is much larger that Ferrier’s Prime. Lucas proved 2^127-1 was prime in 1876, and that held the record of largest prime until Ferrier beat it in 1951.

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**Landry and Aurifeuille**

The mathematician Landry devoted a good part of his life to factoring 2n+1 and finally found the factorization of 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.

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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 2n -1 is prime if it divides the Lucas-Lehmer number Ln where Ln=(Ln-1)2-2 In 1903, Frank Nelson Cole gave a talk where he wrote = 193,707,721 x 761,838,257,287 on the blackboard, then sat down to a standing ovation.

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Palindromic Prime A palindromic prime is a prime that is a palindrome. A pyramid of palindromic primes by G. L. Honaker, Jr. 2 30203 Largest known is (This depends on the base in which the number is written. Mersenne primes are palindromic base 2).

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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. 23, 53, 223, 227,

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Repunit Primes Repunits are positive integers in which all the digits are 1, denoted as R1 = 1, R2=11, etc. Of these, the following are known to be prime:11, , and (2, 19, and 23 digits), R317 ( )/9, and R1,031 ( )/9. In 1999 Dubner discovered that R49081 = ( )/9 was a probable prime, in 2000 Baxter discovered the next repunit probable prime is R86453, and in 2007 Dubner identified R as a probable prime. It is conjectured that there are an infinite number of repunit primes, as illustrated by a graph of repunit primes in a grid of log(log(Rn)) versus n. The points appear to lie very close to a line of constant slope.

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**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

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**Cousin Primes Cousin primes are primes whose difference is 4.**

The first few pairs are {3,7},{7,11},{17,23},{43,47}

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**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}

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Wieferich Prime By Fermat's Little Theorem any prime p divides 2p-1-1. A prime p is a Wieferich prime if p2 divides 2p-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.

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**Named and Notorious Primes**

Early Primes Named Primes Hunting for Primes Harnessing Primes Science News had an article recently about the origins of this spiral.

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**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 Journey through Genius

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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) ) and published in 1808, finding that yet again, Gauss had been there first. How Many Primes.xls Gauss - his motto was “few but ripe” -some have complained that is practices held back mathematics by half a century. He was not interested in material wealth and had Adrien-Marie Legendre, , was cursed with bad timing. He grew up in a wealthy family in France, just before the Revolution wiped out their fortune. He was a tremendous mathematical mind who worked at the same time as one of the greatest mathematicians of all time.

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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

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**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) = 1/1x+1/2x+1/3x+…1/nx+… Explain modular arithmetic as remainder arithmetic Zeta function for s=1 is called harmonic series, when x=2, zeta(x)=(π^2)/6

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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: If you graph the accumulated number of the prime numbers, it is a stairstep graph. Gauss was head of the department when Riemann arrived in Goettingen. His mania for elegance without all the gory details led Riemann to omit messy calculations from his published papers. His housekeeper burned many papers after he died, to clean his room for the next occupant.j What was left was boxed up and eventually stored in the library. Not until Siegel tried reading them in the 1920s did the amazing amount of calculation that Riemann performed come to light.

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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.”

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**Prime Number Sieves Eratothsenes Sieve Excel Sieves Quadratic Sieve**

Number Field Sieve Eratothsenes Sieve can be used to practice multiplication tables. Show the Excel sieves already filled, but not colored.

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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.

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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.

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**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 213,466,917-1, the 39th Mersenne Prime. Five more have been discovered since then.

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Opportunity On September 4, 2006, Dr. Curtis Cooper and Dr. Steven Boone's CMSU team discovered the 44th known Mersenne prime, 232,582,657-1. Edson Smith using GIMPS found 243,112,609-1 (about 12.9 million digits, Aug 08), winning the $100,000 prize from the Electronic Freedom Foundation There is an active community of prime number hunters at primegrid.com. You can join a team, too.

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Prime Generators There are several polynomial functions that generate primes for a while before they start yielding composite numbers. F(x) = x2 + x yields prime number for x < 40.

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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) f2)2 - (e3(e + 2)(a + 1) o2)2 - ((a2 - 1)y x2)2 - (16r2y4(a2 - 1) u2)2 - (((a + u2(u2 - a))2 - 1)(n + 4dy) (x + cu)2)2 - (n + l + v - y)2 - ((a2 - 1)l m2)2 - (ai + k l - i)2 - (p + l(a - n - 1) + b(2an + 2a - n2 - 2n - 2) - m)2 - (q + y(a - p - 1) + s(2ap + 2a + p2 - 2p - 2) - x)2 - (z + pl(a - p) + t(2ap - p2 - 1) - pm)2)

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**Elliptic Curve Factorization**

Faster than the Pollard rho factorization and Pollard p-1 factorization methods. (Wolfram website) Neal Koblitz of the UW was studying elliptic curve factorization when he realized it can be used to encrypt information, also. The technique doesn’t need such big keys or processors, so it is well suited to mobile devices, like cell phones.His company teaches even bankers to play with pointson elliptic curves.

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**Named and Notorious Primes**

Early Primes Named Primes Hunting for Primes Harnessing Primes

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

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**Greatest Common Factors**

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

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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.

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

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

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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. Hugh Montgomery expected the zeroes to be randomly distributed along Riemann’s line, which meant that there had to be clusters every now and then. He didn’t find any clusters. He was shy, but his host thought he ought to meet the great Freeman Dyson. To break the ice, Dyson asked him what he was working on, and he pulled out the graphs from his failed research project.

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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. Demonstrating the existence would confirm the Riemann Hypothesis, but their explicit model only gives an approximation of the energy levels needed. (Science News, 2008)

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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. The last 100 million zeroes were calculated just to win the bet. At $700/hr and 1000 CPU hours, the bottles cost $350,000 each.

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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 (ap-1)=1(mod p). Factoring large numbers is computationally difficult Distribute and discuss RSA

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**Named and Notorious Primes**

Bibliography: Music of the Primes - Marcus du Sautoy, Journey through Genius, William Dunham, Joyce Frost - Joe Frost -

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Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1 Chapter 1 Number Theory and the Real Number System.

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