2 What’s a Prime Number? Lots of definitions out there My Favorite (recursive):“an integer greater than 1, that is not divisible by any smaller primes”Note: The above is equivalent to (but feels less restrictive than) the more standard:“a positive integer greater than 1 that is not divisible by any number other than 1 and itself.”
3 Why Care about Primes?Textbook Answer: Fundamental Theorem of ArithmeticEvery positive integer can be written uniquely as an increasing product of powers of primes- So primes are the “DNA” of integers.Better(?) Answer: Because they’re there!Note if 1 were prime the FTA wouldn’t hold.George Mallory disappeared on Everest in 1924, body found 75 years later. Unknown if he made first ascent.
4 How many?One of the most famous mathematical proofs shows that there are infinitely many.Ancient Greek Mathematician Euclidc. 300 BCFrom “Elements”By contradictionIn a sense, we haven’t made much progress in the 2300 years since this proof.For example, if 2, 3 and 5 were the only primes, we could create a number by multiply them and adding 1. This number, 31, is not divisible by 2, 3 or 5 (because 30 is) so it must be prime. But it wasn’t on our list.
5 Prime producing function? In 1641, Fermat stated that all numbers of the form are prime. Called Fermat primes.f(0) = 3. Prime.f(1) = 5. Prime.f(2) = 17. Prime.f(3) = 257. Prime.f(4) = 65,537. Prime.Convinced?Roughly 100 years later, Euler showed that f(5)= 4,294,967,297 = 641 x 6,700,417. Composite!Today f(4) is still the largest known Fermat primeWe know Fermat numbers from 5 to 32 are compositeThose are big numbers. f(9) > # atoms in universe!We know f(2,747,497) is composite (largest known Fermat composite)We don’t know if there are any more Fermat primesWe don’t know that there aren’t infinitely many Fermat primesWe don’t know if there are infinitely many Fermat compositesScientists estimate atoms in universe between 10^78 and 10^82. 10^81 = (10^3)^27 = 1000^27 = 1024^27 = (2^10)^27 = 2^270 < 2^512 = 2^(2^9) = f(9)
6 Prime producing function? Leonard Euler (1770) noted that many numbers of the form are prime.e(1) = 41. Prime.e(2) = 43. Prime.e(3) = 47. Prime.e(4) = 53. Prime.e(5) = 61. Prime.e(6) = 71. Prime.Convinced?e(7), e(8), e(9), e(10), e(11), e(12), e(13), e(14), e(15), e(16), e(17), e(18), e(19), e(20), e(21), e(22), e(23), e(24), e(25), e(26), e(27), e(28), e(29), e(30), e(31), e(32), e(33), e(34), e(35), e(36), e(37), e(38), e(39), e(40) all prime.e(41) = 41* = 41 ( ) = 41 x 41. Composite!Can show that no polynomial function can produce only primes.Interestingly, any linear function (of the from an + b) produces infinitely many primes, if a and b are themselves prime.F(constant term) = constant term * (something), if constant not 1.Actually, only need a and b relatively prime
7 Prime producing function In short, we don’t know of one.In 1947 Mills proved that is always prime, for some A.Unfortunately we don’t know what A isWe don’t even know if A is rational or irrationalNot aesthetically pleasing to use floor functionBottom line is that we don’t know of any prime producing functionbut we know there is oneHopefully a prettier one than the aboveIf RH is true, A = … And mills primes are 2; 11; 1,361; 2,521,008,887; 16,022,236,204,009,818,131,831,320,183Reimann Hypothesis proposed in 1859 is considered by many the greatest unsolved problem in math. One of 7 unsolved Miillenium problem.
8 Mersenne primes Marin Mersenne, a French Monk born in 1588 The nth Mersenne number isSeveral Mersenne numbers are prime m(2)=3, m(3)=7, etc.m(5), m(7), also primem(composite) = compositeMathematicians once thought m(prime)=primeWrong!Mersenne numbers have algebraic properties that are useful in determining primalityDifference of squares, for exampleM(100) composite asBorn in a peasant familyM(11) composite
9 Largest known prime A game that will never end Some think that size of largest prime is a good measure of society’s knowledgeImplies exponential growth of knowledgeLots of early claims of large primesMany were wrongEuler (1772) proved primeIn 1876 m(127) shown to be primeRecord lasted until 1951Largest ever without computers (39 digits)M(67) removed from list in 1903 in famous hour long “talk”M(67) =147,573,952,589,676,412,927 = 193,707,721 x 761,838,257,287M(67) =147,573,952,589,676,412,927 = 193,707,721 x 761,838,257,287Frank Cole found this (with no calculator)
11 Largest known primes Gimps – Great Internet Mersenne Prime Search Note log scale on y axis
12 Random? Primes appear to be scattered at random. No (known) way to generate themNo (known) way to (easily) tell if a number is primeSo are they scattered randomly?Is there a pattern that we’re not smart enough to see?YesHypothesized by Brian Stonelake (2013)
19 Less Arbitrary Visual Representations Variant of Sach’s SpiralDot size determined by unique prime factors
20 How little we knowPrime numbers, the DNA of all numbers, are remarkably mysterious.We can’t generate themWe don’t have a method for recognizing themThey don’t appear random, but we can’t describe their patternWhat can we say about them?
21 Distribution of Primes Less thanNumber of primesProbability of a prime10440%1002525%1,00016817%10,0001,22912%100,0009,5929.6%1,000,00078,4897.9%1,000,000,00050,847,5345.1%1,000,000,000,00037,609,912,0183.8%1,000,000,000,000,00029,844,570,422,6693.0%Probability seems to be decreasing. Is there some sort of pattern?
23 Prime number theorem (PNT) PNT says that primes become less common among large numbers, and do so in a predictable fashion.Approximates the number of primes less than n as L(n) = n/ln(n).The nth prime number is approximately n*ln(n)Also says that is an approximation of primes less than n.This approximation is closer, sooner.Actually, it says the limit of the ratio of n/ln(n) and pi(n) goes to 1 as n goes to infinity.Proven in 19th century, “simple” proof in 1980.
24 Prime Number Theorem n π(n) L(n) Li(n) π(n) / L(n) π(n) / Li(n) 10 4 4.36.21002522301,00016814517810,0001,2291,0861,246100,0009,5928,6869,63010^678,49872,38278,62810^7664,579620,421664,91810^85,761,4555,428,6815,762,20910^950,847,53448,254,94250,849,23510^10455,052,511434,294,482455,055,61510^114,118,054,8133,948,131,6544,118,066,40110^1237,607,912,01836,191,206,82537,607,950,28110^13346,065,536,839334,072,678,387346,065,645,81010^143,204,941,750,8023,102,103,442,1663,204,942,065,69210^1529,844,570,422,66928,952,965,460,21729,844,571,475,28810^16279,238,341,033,925271,434,051,189,532279,238,344,248,55710^172,623,557,157,654,2302,554,673,422,960,3002,623,557,165,610,820Skewes first proved that pi(x) < li(x) at 10^10^10^963. An absurd number called skewes’ number. Recall there are approximately 10^81 atoms in universe.Since improved to e^ = 10^316Infinitely many places where sign of difference changes
25 Prime number theoremIF RH is true, we know that this is the best possible approximation. Without it we just have the asypmtotic behavior.We can formally show the intuitive result that primes are less common among larger numbers
26 A giant’s walk to infinity PNT says large numbers are less likely to be primeIntuitively, there are more primes that could divide itSo primes get more and more “spread out”Imagine walking on a number line, where only primes are stepsHow far could you get?I can jump 5 units, where do I get stuck?How far would I need to be able to jump to get to 100?Could anyone get to infinity?----- Meeting Notes (10/15/14 20:32) -----(I get stuck at 23)
27 Prime GapsThe difference between two consecutive primes is called the prime gap. The first few prime gaps are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, …PNT suggests prime gaps get largerBut there’s infinitely many primesLargest prime gap?We can create arbitrarily large prime gaps, by following the following examplePrime gap of g = 14Multiply all primes less than or equal to g+2. Call that product b.b = 2 x 3 x 5 x 7 x 11 x 13 = 30,03030,032 to 30,046 can’t contain any primesNote there’s also no primes between 113 and 127So we can (easily) find sequences of arbitrarily length that contain no primes at all!Even a giant can’t get to infinity!
28 Twin primes 2 and 3 are the only primes with gap 1 Many have gap 2; called twin primes(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139)Infinitely many?Nobody knows (called Twin Prime Conjecture)Dates back to at least 1849In March 2013, Zhang showed that there are infinitely many prime “brothers” with gap of some (unknown) number less than 7 millionIn July the gap bound was reduced to 5,414Most believe TPC true
29 Convergent/Divergent Harmonic series divergesSquares of Harmonic series convergesToCalled Basel Problem (1644), solved by… EulerWhat about reciprocals of primes?Are they “frequent enough” to diverge?Yes (Euler)Shocking?What about reciprocals of twin primes?They converge (to Brun’s constant)We don’t know the constant, it’s very close to
30 Gaussian Primes Extending the concept of “prime” to complex numbers Gaussian integers are complex numbers of the form a+bi where a and b are integers2+i is Gaussian prime because no two (non-trivial) Gaussian integers have 2+i as their productNote 5 not Gaussian prime as (2+i)(2-i) = 5a+bi Gaussian prime if and only if:a = 0 and is prime andb = 0 and is prime andis prime(A+Bi)(C+Di)=(AC-BD) + (AC+BD)I
33 Gaussian PrimesIs there a giant that could walk on Gaussian primes to infinity?Nobody knowsBest we can do is say that a giant that can’t jump 6 couldn’t do it!We know there are “moats” of arbitrary size around Gaussian primes, but that doesn’t helpInfinitely many?Yes. In fact, Infinitely many that are ordinary primes.Largest known (absolute value) isReal and imaginary parts have 181,189 digits!Mersenne-ish
34 Goldbach Conjecture Considers sums of primes Every even integer greater than 2 can be expressed as the sum of two primes.One of the oldest unsolved problems in mathProposed (to Euler) in 1742True for all even integers up to 4,000,000,000,000,000,000Generally thought to be true, but who knows?Is it possible that it’s true but unprovable?An author offered $1,000,000 prize for proof or counterexample in 2002
36 Goldbach ConjectureNumber of ways two primes sum to each even integer up to 1,000
37 Goldbach ConjectureNumber of ways two primes sum to each even integer up to 1,000,000
38 Riemann Hypothesis (RH) Considered by most the most important problem in mathZeta function isRH says that the (non-trivial) zeros of the Zeta function all have real part ½.Known to be true for the first 10,000,000,000,000 zerosIf RH is true, there are TONS of implications.A major one tells us Li(x) is the best approximation of prime distribution, and gives error bounds on it.Minor ones:Reduces Skewes number from 10^10^10^963 to 10^10^10^34“A” in Mills prime producing function is approximately …Trivial zeros at x=-2n“Prime Obsession” a good read.