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Named and Notorious Primes
Joe Frost
University of Washington Computing and
Communications
Joyce Frost
Lake Washington High School
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
A History and Exploration of Prime Numbers
Dedicated to Royal Penewell -1923-2008
Math Teacher and Prime Enthusiast
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Named and Notorious Primes
• Early Primes
• Named Primes
• Hunting for Primes
• Harnessing Primes
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.
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Eratosthenes of Cyrene276-194 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.
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.
Pierre de Fermat1601-1665
• 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 (ap-1-1).
• Mentioned in a letter in 1640 with no proof, proved by Euler in 1736
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 ap-a.”
Carl Friedrich Gauss1777-1855
• 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.
Bernhard Riemann1826-1866
• 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) =∏(ns)/(ns -1) (n=2,3,5,7,11,…)
Named and Notorious Primes
• Early Primes
• Named Primes
• Hunting for Primes
• Harnessing Primes
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.
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.
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, 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: Wn=n*2n -1. These are then the "Cullen primes of the second kind".
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. €
n ∈ 0,1,2,3,4,?{ }
€
22n
+1
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.
Ferrier’s Prime
• Ferrier’s Prime is the largest prime found before electronic calculators. Ferrier’s Prime = 1/17(2148+1) = 20988936657440586486151264256610222593863921
Fibonacci Prime
• A Fibonacci prime is a Fibonacci number that is prime.
• 1,1,2,3,5,8,13,21,34,55,89,144…
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.
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.
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.
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.
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).
Landry and Aurifeuille
• The mathematician Landry devoted a good part of his life to factoring 2n+1 and finally found the factorization of 258+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.
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
Palindromic Prime• A palindromic prime is a prime that is a palindrome. A
pyramid of palindromic primes by G. L. Honaker, Jr.2
30203133020331
171330203317112171330203317121
1512171330203317121511815121713302033171215181
16181512171330203317121518161331618151217133020331712151816133
• Largest known is 105901146541059011
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.
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, 1111111111111111111, and 11111111111111111111111 (2, 19, and 23 digits), R317 (10317-1)/9, and R1,031 (101031-1)/9.
• In 1999 Dubner discovered that R49081 = (1049081-1)/9 was a probable prime, in 2000 Baxter discovered the next repunit probable prime is R86453, and in 2007 Dubner identified R109297 as a probable prime.
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
Cousin Primes
• Cousin primes are primes whose difference is 4.
• The first few pairs are {3,7},{7,11},{17,23},{43,47}
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}
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.
Named and Notorious Primes
• Early Primes
• Named Primes
• Hunting for Primes
• Harnessing Primes
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
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
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
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+…
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:
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.”
Prime Number Sieves
• Eratothsenes Sieve
• Excel Sieves
• Quadratic Sieve
• Number Field Sieve
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.
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.
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.
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
Prime Generators
• There are several polynomial functions that generate primes for a while before they start yielding composite numbers.
• F(x) = x2 + x + 41 yields prime number for x < 40.
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 - f2)2 - (e3(e + 2)(a + 1)2 + 1 - o2)2 - ((a2 - 1)y2 + 1 - x2)2 - (16r2y4(a2 - 1) + 1 - u2)2 - (((a + u2(u2 - a))2 - 1)(n + 4dy)2 + 1 - (x + cu)2)2 - (n + l + v - y)2 - ((a2 - 1)l2 + 1 - m2)2 - (ai + k + 1 - 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)
Elliptic Curve Factorization
Faster than the Pollard rho factorization and Pollard p-1 factorization methods. (Wolfram website)
Named and Notorious Primes
• Early Primes
• Named Primes
• Hunting for Primes
• Harnessing Primes
Prime Factorization
Every number can be expressed as a unique product of prime numbers.
Example: 450 = 2*3*3*5*5
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*55*5*5
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.
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.
Communication
In Carl Sagan’s novel Contact, aliens send a series of prime numbers to show intelligence behind radio transmissions
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.
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.
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.
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
