Ulam's Rose or Prime Number Spiral

Prime Numbers seem to occur randomly along the Number Sequence but Ulam found a mysterious spiral pattern...!

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8.2 Ulam's Rose or Prime Number Spiral

— There's Something About the Number Sequence —

Prime Numbers (2, 3, 5, 7, 11, 13, 17,...)

There are many kinds of numbers and all different kinds come down to the same thing: they are amounts of something before the something is added. In a weird and far sought way, all numbers are equal, but some numbers are more equal than others. Those are the prime numbers, and primes are of royalty.

Prime numbers are numbers that can only be divided by themselves and by one, which in both cases comes down to not dividing at all.

Ulam's Rose
Ulam's Rose or
Prime Number Spiral

Primes are diamond numbers. They're rock hard and can't be tampered with. Primes are the atoms of the number sequence and all other numbers are built from primes. And that's why we call them composite numbers. Composite numbers are molecules. Primes are atoms.

And the number sequence is so arranged that some molecules (like 4) are a lot smaller than some atoms (like 21398269-1). Nature obviously doesn't work like that, although some molecules (like H2O) are a lot smaller than some atoms (like uranium). Still, molecule-atom ratios like 4 / 21398269-1 do not occur in nature. Because nature isn't based on an infinite string of atoms. The number sequence is. In fact, it is provable that between a certain number N and that same number times two (2N) at least 1 prime number has to exist. (Click here for a mathematical proof of this).

Prime numbers have fascinated mathematicians for as long as the existence of primes in the context of the number sequence has been discovered. And the big question soon became: why do prime numbers occur at such inconsistent intervals? Could there be one single formula that predicts all prime numbers, or something that can be said that is true for all prime numbers? Is there any kind of regularity in the appearance of primes?

No one could supply the world with an answer and primes where believed to occur randomly. And that's a very big word because randomness is deeply akin to the freedom upon which nature is based. True randomness can not be predicted because there is no deterministic force behind it. With her prime numbers, the number sequence seemed to prove that numbers can indeed be used to express randomness, hence to describe nature, hence cover the Truth. And the excitement about primes flared up even more in the wake of boredom of a devoted 20th century math-magician named Stanislaw Ulam.

Stan the Man.

Stanislaw Ulam
Stanislaw Ulam

Stanislaw Ulam was attending some boring meeting, and to divert himself somewhat he began to scribble on a piece of paper. If anything other than numbers had been his forte he might have doodled flowers and petals or little faces and aliens descending from the sky, but Stan's mind was a number mind and Stan drew numbers.

He put down the number 1 as the bright shining center of a universe of numbers that Big Banged outwardly in a spiral:

737475767778798081
724344454647484950
714221222324252651
704120789102752
694019612112853
683918543122954
673817161514133055
663736353433323156
656463626160595857

Much to his amazement the prime numbers appeared to gravitate towards diagonal lines emanating from the central 1. Yet there was no apparent rule that forced all prime numbers upon a diagonal line like that. Most of them sat on or in the vicinity of a diagonal, but some obviously didn't. Ulam ran home and expanded the spiral to cover a much larger portion of the number sequence. The strange pattern persisted!

Primes had a tendency to occur in clusters and all clusters tended to make a beautiful image that could not be predicted. Click on the thumbnail in the upper right corner of this page. It looks like something out of nature but in fact it's the prime numbers from 1 to 262,144. Quite besotting in a Tyler sort of way. Like water molecules huddle together to make a snow flake according to some basic design, prime numbers huddle together to make the Ulam Rose.

The arythmophilic world responded in awe. There's true, unpredictable randomness in the number sequence! Numbers are as beautiful as nature! And up to this day every book on popular mathematics uses the word random in direct relation to prime number distribution.

Yeah, well. If we're willing to sacrifice that which we think we see in order to investigate what is really there, we may put our awe to rest.

Step up to the next page and look at the Ulam Rose some more, and more careful:
Ulam's Rose? Get a Grip!

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