Optimus/Pessimus Prime

History is riddled with stories of great mathematical minds.  Individuals that saw patterns no normal eye could see, correlations no sane mind could reason.

Carl Friedrich Gauss, for example, is rumored to have been such a prodigy that by age three he was already correcting his father's arithmetic.  My personal favorite Gaussian anecdote happened during his primary school years.  A teacher by the name of J.G. Büttner gave his class what was supposed to be a long and drawn out arithmetic exercie, adding all the numbers 1 through 100.  Before Büttner had finished writing the problem on the board, Gauss had scribbled 5050 on his little chalk board, flung it on his teacher's desk, and exclaimed Ligget se, "There it lies" in peasant tongue.  To see how he came to this answer click here.

Even philosophers of old offered powerful insight into mathematics, my favorite of which was René Descartes.  Although not intentially meant to be a mathematical proof, I take the statement Cogito ergo sum: "I am, I exist", or more commonly translated as "I think therefore I am."; as the root to all mathematical proofs.  If I were asked "How do you know 1 + 1 = 2?", I would respond "Because 1 exists."  If I were then asked "How do you know 1 exists?", I would respond "Because I exist."  And, if I were finally asked "How do you know that you exist?", I would answer "Cogito ergo sum".

The sad/happy part in all of this history of genius is that a simple formula, or proof, for one of the most fundamental mathematical concepts eludes us to this day.  I am speaking of course of Prime Numbers.

A prime number is simply a number that cannot be divided by anything besides itself and the number 1.  There are several intruiging characteristics about primes:

1. Every even integer greater than 2 can be expressed as the sum of two primes, also known as Goldbach's conjecture.  Although every tested even number follows this conjecture, there is still no proof explaining why this is the case.
2. There is no known pattern that predicts prime numbers.
3. There is no equation that can identify specific prime numbers, at least none that don't require foreknowledge of existing primes.
4. In spite of 2 & 3, prime numbers appear to follow a visible pattern that can be seen in many visual representations:

Binary Bits:

Prime Base Plot

Number Spirals:

Ulam Spiral

Sack's Spiral

The best quote I've run across explaining the behavior of prime numbers came from Don Zagier:

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.

There have been prizes numbered in the millions, large groups of genius level professors and incredibly intelligent computer systems that have tried (and failed) to find a simple pattern, proof or formula predicting prime numbers.

Even with the aid of computers and complex algorithms, only 33 new prime numbers have been discovered since the 1950s.

Herein lies the dilemma for math geeks and enthusiasts alike.  If the likes of Gauss, Euclid and countless others from the beginning of time to today could not unlock the mystery of the primes, is the mystery solveable?  Even if it is, can it be done in a lifetime?  Does it take a prodigy, or can a stroke of genius or moment of clarity provide the solution?  Is the answer an elegant and simple one that requires a paradigm shift, or is the very nature of prime numbers the fact that they cannot be expressed as a pattern.

I personally believe there is a simple and elegant proof and equation that explain and predict all the prime numbers from negative infinity to positive infinity.  The key may lie somewhere in chaos theory, probability, fractals, the discovery of a new constant, the creation of a new imaginary number, theoretical physics or even in nature.

I like to think of myself as an optimist when it comes to matters of science, math and human potential.  I know a simple and elegant proof for primes must exist because all the laws of math and nature we have discovered so far have been simple and elegant.  I know math and nature exist because I can conceive of them and understand them to a point.  And, I know I exist because I think therefore I am.