1. Prime Numbers
By Brian Stonelake

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 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.

4. 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.

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 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)

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 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 = … 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.

8. 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

9. 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)

10. Largest known primes
100k prize for 2008 one.

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 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)

13. First 100 primes
But base 10 is arbitrary.

14. Less Arbitrary Visual Representations
Called Ulam spiral

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

16. Less Arbitrary Visual Representations
Archimedean Spiral

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

18. Less Arbitrary Visual Representations

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

20. 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?

21. 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?

22. Distribution of Primes

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
4.3
6.2
100 25 22 30
1,000
168
145
178
10,000
1,229
1,086
1,246
100,000
9,592
8,686
9,630
10^6
78,498
72,382
78,628
10^7
664,579
620,421
664,918
10^8
5,761,455
5,428,681
5,762,209
10^9
50,847,534
48,254,942
50,849,235
10^10
455,052,511
434,294,482
455,055,615
10^11
4,118,054,813
3,948,131,654
4,118,066,401
10^12
37,607,912,018
36,191,206,825
37,607,950,281
10^13
346,065,536,839
334,072,678,387
346,065,645,810
10^14
3,204,941,750,802
3,102,103,442,166
3,204,942,065,692
10^15
29,844,570,422,669
28,952,965,460,217
29,844,571,475,288
10^16
279,238,341,033,925
271,434,051,189,532
279,238,344,248,557
10^17
2,623,557,157,654,230
2,554,673,422,960,300
2,623,557,165,610,820
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^ = 10^316
Infinitely many places where sign of difference changes

25. 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

26. 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)

27. 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!

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 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

29. 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

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 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

31. Gaussian Primes

32. Gaussian Primes

33. 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

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 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

35. Goldbach Conjecture

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

37. Goldbach Conjecture
Number 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 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 …
Trivial zeros at x=-2n
“Prime Obsession” a good read.

39. The End
Questions? Comments?