Prime Numbers By Brian Stonelake

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.”

Why Care about Primes? Textbook Answer: Fundamental Theorem of Arithmetic Every 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.

How many? One of the most famous mathematical proofs shows that there are infinitely many. Ancient Greek Mathematician Euclid c. 300 BC From “Elements” By contradiction In 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.

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 prime We know Fermat numbers from 5 to 32 are composite Those 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 primes We don’t know that there aren’t infinitely many Fermat primes We don’t know if there are infinitely many Fermat composites Scientists 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)

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 + 41 = 41 (41 - 1 + 1) = 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

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 is We don’t even know if A is rational or irrational Not aesthetically pleasing to use floor function Bottom line is that we don’t know of any prime producing function but we know there is one Hopefully a prettier one than the above If RH is true, A = 1.306377883… And mills primes are 2; 11; 1,361; 2,521,008,887; 16,022,236,204,009,818,131,831,320,183 Reimann Hypothesis proposed in 1859 is considered by many the greatest unsolved problem in math. One of 7 unsolved Miillenium problem.

Mersenne primes Marin Mersenne, a French Monk born in 1588 The nth Mersenne number is Several Mersenne numbers are prime m(2)=3, m(3)=7, etc. m(5), m(7), also prime m(composite) = composite Mathematicians once thought m(prime)=prime Wrong! Mersenne numbers have algebraic properties that are useful in determining primality Difference of squares, for example M(100) composite as Born in a peasant family M(11) composite

Largest known prime A game that will never end Some think that size of largest prime is a good measure of society’s knowledge Implies exponential growth of knowledge Lots of early claims of large primes Many were wrong Euler (1772) proved prime In 1876 m(127) shown to be prime Record lasted until 1951 Largest 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 761,838,257,287 M(67) =147,573,952,589,676,412,927 = 193,707,721 x 761,838,257,287 Frank Cole found this (with no calculator)

Largest known primes 100k prize for 2008 one.

Largest known primes Gimps – Great Internet Mersenne Prime Search Note log scale on y axis

Random? Primes appear to be scattered at random. No (known) way to generate them No (known) way to (easily) tell if a number is prime So are they scattered randomly? Is there a pattern that we’re not smart enough to see? Yes Hypothesized by Brian Stonelake (2013)

First 100 primes But base 10 is arbitrary.

Less Arbitrary Visual Representations Called Ulam spiral

Less Arbitrary Visual Representations Ulam’s Spiral Random “white noise”

Less Arbitrary Visual Representations Archimedean Spiral

Less Arbitrary Visual Representations Sack’s Spiral: Uses Archimedean Spiral

Less Arbitrary Visual Representations

Less Arbitrary Visual Representations Variant of Sach’s Spiral Dot size determined by unique prime factors

How little we know Prime numbers, the DNA of all numbers, are remarkably mysterious. We can’t generate them We don’t have a method for recognizing them They don’t appear random, but we can’t describe their pattern What can we say about them?

Distribution of Primes Less than Number of primes Probability of a prime 10 4 40% 100 25 25% 1,000 168 17% 10,000 1,229 12% 100,000 9,592 9.6% 1,000,000 78,489 7.9% 1,000,000,000 50,847,534 5.1% 1,000,000,000,000 37,609,912,018 3.8% 1,000,000,000,000,000 29,844,570,422,669 3.0% Probability seems to be decreasing. Is there some sort of pattern?

Distribution of Primes

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.

Prime Number Theorem n π(n) L(n) Li(n) π(n) / L(n) π(n) / Li(n) 10 4 4.3 6.2 0.92103 0.64516 100 25 22 30 1.15129 0.83056 1,000 168 145 178 1.16050 0.94382 10,000 1,229 1,086 1,246 1.13195 0.98636 100,000 9,592 8,686 9,630 1.10432 0.99605 10^6 78,498 72,382 78,628 1.08449 0.99835 10^7 664,579 620,421 664,918 1.07117 0.99949 10^8 5,761,455 5,428,681 5,762,209 1.06130 0.99987 10^9 50,847,534 48,254,942 50,849,235 1.05373 0.99997 10^10 455,052,511 434,294,482 455,055,615 1.04780 0.99999 10^11 4,118,054,813 3,948,131,654 4,118,066,401 1.04304 1.00000 10^12 37,607,912,018 36,191,206,825 37,607,950,281 1.03915 10^13 346,065,536,839 334,072,678,387 346,065,645,810 1.03590 10^14 3,204,941,750,802 3,102,103,442,166 3,204,942,065,692 1.03315 10^15 29,844,570,422,669 28,952,965,460,217 29,844,571,475,288 1.03079 10^16 279,238,341,033,925 271,434,051,189,532 279,238,344,248,557 1.02875 10^17 2,623,557,157,654,230 2,554,673,422,960,300 2,623,557,165,610,820 1.02696 Skewes 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^727.95133 = 10^316 Infinitely many places where sign of difference changes

Prime number theorem IF 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

A giant’s walk to infinity PNT says large numbers are less likely to be prime Intuitively, there are more primes that could divide it So primes get more and more “spread out” Imagine walking on a number line, where only primes are steps How 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)

Prime Gaps The 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 larger But there’s infinitely many primes Largest prime gap? We can create arbitrarily large prime gaps, by following the following example Prime gap of g = 14 Multiply 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 = 30,030 30,032 to 30,046 can’t contain any primes Note there’s also no primes between 113 and 127 So we can (easily) find sequences of arbitrarily length that contain no primes at all! Even a giant can’t get to infinity!

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 1849 In March 2013, Zhang showed that there are infinitely many prime “brothers” with gap of some (unknown) number less than 7 million In July the gap bound was reduced to 5,414 Most believe TPC true

Convergent/Divergent Harmonic series diverges Squares of Harmonic series converges To Called Basel Problem (1644), solved by… Euler What 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 1.830484424658

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 integers 2+i is Gaussian prime because no two (non-trivial) Gaussian integers have 2+i as their product Note 5 not Gaussian prime as (2+i)(2-i) = 5 a+bi Gaussian prime if and only if: a = 0 and is prime and b = 0 and is prime and is prime (A+Bi)(C+Di)=(AC-BD) + (AC+BD)I

Gaussian Primes

Gaussian Primes

Gaussian Primes Is there a giant that could walk on Gaussian primes to infinity? Nobody knows Best 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 help Infinitely many? Yes. In fact, Infinitely many that are ordinary primes. Largest known (absolute value) is Real and imaginary parts have 181,189 digits! Mersenne-ish

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 math Proposed (to Euler) in 1742 True for all even integers up to 4,000,000,000,000,000,000 Generally 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

Goldbach Conjecture

Goldbach Conjecture Number of ways two primes sum to each even integer up to 1,000

Goldbach Conjecture Number of ways two primes sum to each even integer up to 1,000,000

Riemann Hypothesis (RH) Considered by most the most important problem in math Zeta function is RH 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 zeros If 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 1.306377883863080690486144926… Trivial zeros at x=-2n “Prime Obsession” a good read.

The End Questions? Comments?