Prime Numbers By Brian Stonelake

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 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 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 irrationalNot aesthetically pleasing to use floor functionBottom line is that we don’t know of any prime producing function but we know there is oneHopefully a prettier one than the above

Mersenne primes

Marin

Mersenne

, a French Monk born in 1588

The n

th

Mersenne

number is

Several 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 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 Primes

Less

than

Number of primes

Probability of a prime

10

4

40%

100

2525%1,00016817%

10,0001,22912%100,0009,592

9.6%

1,000,000

78,4897.9%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?

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 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!

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

+

b

i

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 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

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 1.306377883863080690486144926…

The End

Questions?

Comments?