Stuff I Dont Know Unsolved Problems in Mathematics

Stuff I Don’t Know Unsolved Problems in Mathematics

Collatz Conjecture • “The Simplest, Impossible Problem” • Pick a number (positive integer) – If it’s even, cut it in half – If it’s odd, triple it then add one – Repeat • Will you eventually end up at one? – If not, you solved a problem that has baffled mathematicians for almost 80 years. • “Mathematics might not be ready for such problems” - Erdos

Collatz Conjecture • If you picked 27, it took you 111 steps, and you got as high as 9232, before getting to 1. • If you picked 63, 728, 127 it took you 949 steps • Confirmed up to 2 to the 60 th power – Aka 1. 1 Quintillion – Can you confirm it for any numbers bigger than that?

Careful with Empirical “Proofs”

Collatz Stopping Time

CST Histogram • Math 243 fans? Who loves histograms? • Stopping times for 1 to 100 million

Collatz Trajectory

Collatz in Reverse

Collatz Map, n = 20

Collatz Map, n = 30

Collatz Map, n = 50

Collatz Map, n = 100

Collatz Map, n = 500

Collatz Map, n = 10, 000

Collatz 3 D Map, n = 10, 000

Collatz Fractal

xkcd’s Take on Collatz

Chromatic Number of the Plane • Aka the Hadwiger – Nelson Problem • Chromatic number: minimum colors needed, to paint vertices, so no two adjascent vertices share a color. • Quiz? • Quiz!

Chromatic Warm-up Determine the Chromatic number of each of the 6 graphs below.

Chromatic Challenge

It was 3!

Chromatic Challenge II

Not 3!

Similar to 4 -color theorem • With four colors, you can color regions in any map, without having adjacent regions the same color

4 -Color Theorem Example

Infinite Map? • What if every point in the plane were connected to all points a distance of one from it? • How many colors to “color the plane”?

Not less than 4!

Not more than 7!

Open Quesitons • So we know the chromatic number of the plane is at least 4 and at most 7. – So what is it? – Nobody knows? – I guess 4… people are good at finding counterexamples. • What about the chromatic number of 3 space? – At least 6 and at most 15, but no clue which.

“Most Unsolved” Question? • Prime numbers! • DNA of integers, but we can’t predict them • Arbitrarily large gaps between them, but they never run out • Infinitely many, but no easy way to find or verify them • Make up a prime question, it’s probably open.

Prime producing function • In short, we don’t know of one. – In 1947 Mills proved that prime, for some A. is always • 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

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 Archimedean Spiral

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

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

Prime producing function? • In 1641, Fermat stated that all numbers of the form Called Fermat primes. – f(0) = 3. Prime. – f(1) = 5. Prime. – f(2) = 17. Prime. – f(3) = 257. Prime. – f(4) = 65, 537. Prime. are 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 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

Other Open Prime Questions • Infinitely many: – Twin primes? – Palindrome primes? – Sexy primes? – Fibbonacci primes? – Circular primes? – Permutable primes – Repunit primes? – Repunit primes in other bases?

Goldbach Conjecture Can any even number be expressed as the sum of two primes?

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

Gaussian Primes • Same idea, but extended to complex numbers. • Turns out 2 + i and 2 – i are Gaussian prime. – So is 3. – What about 5? • Not as mysterious as you might hope… – a + bi prime iff prime, for example • But they look cool on a graph!

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!

Most Important Unsolved Problem • “The non-trivial zeros of the Reimann zeta function , all have real part ½. ” • Really the analytic continuation of the above • Millennium Prize Problem • True for

Sieve of Erastothenes

Amazingly, !

Enough Algebra, Pictures Please

Last Picture Slides available at http: //webpages. sou. edu/~stonelakb/math/