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!

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.

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

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.

• Convinced?– 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.

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

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

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

Largest known primes

Largest known primes

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

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 PrimesLess 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.

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 1.00000

10^13 346,065,536,839 334,072,678,387 346,065,645,810 1.03590 1.00000

10^14 3,204,941,750,802 3,102,103,442,166 3,204,942,065,692 1.03315 1.00000

10^15 29,844,570,422,669 28,952,965,460,217 29,844,571,475,288 1.03079 1.00000

10^16 279,238,341,033,925 271,434,051,189,532 279,238,344,248,557 1.02875 1.00000

10^17 2,623,557,157,654,230 2,554,673,422,960,300 2,623,557,165,610,820 1.02696 1.00000

Prime number theorem

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?

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

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…

The End

Questions? Comments?