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Ulam's Rose? Get A Grip!

Source: http://www.abarim-publications.com/ulamgohome.html

8.3 Ulam's Rose? Get A Grip!

— Rose? What Rose? —

A rose by any other name...

Alright folks! This has been going on too long. Let's get real about the Ulam Prime conundrum. First of all, let's wonder what exactly the Ulam Spiral is. Well, to start with, the Ulam Spiral is not a spiral but isolated segments of the number sequence that get larger 8 numbers per segment:

 Number segment Numbers a → b Amount of numbers
 heart 1 1
 1 2 → 9 1x8
 2 10 → 25 2x8
 3 26 → 50 3x8
 4 51 → 83 4x8
 etc etc etc

The Amazing Abarim Publications Pillar

 2 3 . 5 . 7
... 11. 13
... 17. 19
... 23. *
... 29. 31
... *. 37
... 41. 43
... 47. *
... 53. *
... 59. 61
... *. 67
... 71. 73
... etc. etc

But why would we want to write down the number sequence in segments that get 8 numbers larger every time? Why not write it down in segments of 6 that stay 6? Behold, the Amazing Abarim Publications Pillar in which all primes numbers larger than 3 occur along two perfectly straight lines!

As you can see, in the Amazing Abarim Publications Pillar, all primes after 3 are neatly sorted in two columns. There's never one out. All primes mankind is ever going to find are in one of these two columns: 6a+1 or 6a-1 (a being any number).

Most of the time, prime numbers find themselves adrift in an expanding ocean of composites, but sometimes one of those 6a+ or -1 stay in tact (like 41-43 or 71-73). Then they form a so-called prime-twin. Many pop-math books will try to make you believe that no one knows why prime-twins exist, but now you know better.

But why are some numbers in the two pillars crossed out and turned into a little star? It's when a so-called iso-factor crosses them out. Do yourself a favor. Print this page and then write in the Amazing Abarim Publications Pillar all numbers that are divisible by 5. You will find that they too are neatly lined up, except that the line is not vertical but diagonal.

Imagine the Amazing Abarim Publications Pillar to be rolled up to a tube so that the number sequence becomes a spiral swirling down that tube. You will find that all iso-factors (lines that connect all numbers divisible by some other number) are spirals too, and swirl along the same tube.

A rose without thorns

Let's do another clever trick. Let's pretend that the two prime columns of the Amazing Abarim Pillar are not interrupted by iso-factor victims, but perfectly solid with prime numbers. We can do that because for as long as the number sequence runs, its possible prime numbers occur in the same rhythmic cadence, namely 'number number number possible-prime number possible-prime'. Like so ('P' is a possible prime):

1 2 3 * P * P * * * P * P * * * P * P * * * P * P * * * P * P

It's the basic, underlying pattern of the number sequence. Now watch what happens if we coil this basic pattern of the number sequence up like Ulam did:

 . P . . . P . P . . . P . P . . . P . P . . . P .
 P . . . P . P . . . P . P . . . P . P . . . P . P
 . P . P . . . P . P . . . P . P . . . P . P . . .
 . . P . P . . . P . P . . . P . P . . . P . P . .
 . P . P . . . P . P . . . P . P . . . P . P . P .
 P . . . P . P . . . P . P . . . P . P . . . . . P
 . . . . . P . P . . . P . P . . . P . P . . . P .
 P . P . P . P . . . P . P . . . P . P . P . P . P
 . P . P . . . P . P . . . P . P . . . . . P . . .
 . . P . . . . . P P . . . P . P . . . . P . P . .
 . P . P . P . P . . . . P . P . . P . P . P . P .
 P . . . P . P . . P . P . . . . . . P . . . . . P
 . . . . . P . . . . P .  1 2 P . P P . P . . . P .
 P . P . P . P . P . . P . 3 . P . . P . P . P . P
 . P . P . . . P . . P . . . P . P . . . . P . . .
 . . P . . . . . P P . P . . . P . . . . P . P . .
 . P . P . P . P . . . P . P . . . P . P . P . P .
 P . . . P . P . . . P . P . . . P . P . . . . . P
 . . . . . P . P . . . P . P . . . P . P . . . P .
 P . P . P . . . P . P . . . P . P . . . P . P . P
 . P . P . . . P . P . . . P . P . . . P . P . . .
 . . P . P . . . P . P . . . P . P . . . P . P . .
 . P . . . P . P . . . P . P . . . P . P . . . P .
 P . . . P . P . . . P . P . . . P . P . . . P . P
 . P . . . P . P . . . P . P . . . P . P . . . P .

Now realize that although the prime number density thins out gradually, they still appear at more or less regular intervals, et voila: you have yourself a magical hoax.

The number sequence cycles. The most basic cycle is obviously the odd-even-odd-even cycle, which lasts 2 numbers and repeats ad infinitum. All prime numbers (except 2) occur in the odd bunch. Write down all numbers in a two-column pillar, and you'll see that all prime numbers are in the same column. Thus we've identified 50% of all numbers to be definitely not prime.

The next cycle is the one we displayed above, which lasts 6 numbers, and repeats add infinitum. All prime numbers after 3 occur in the fourth or sixth column. Thus we've identified 66% of all numbers to be definitely not prime

The next cycle is 2x3x5=30 long and repeats for ever; all primes occur in a limited number of columns. Next comes 2x3x5x7, after that 2x3x5x7x11, then 2x3x5x7x11x13, and so on until the cows come home.

Numbers: a timeless, spaceless and awesome stage for fine fiction

Most of us like to believe that the number sequence begins with 1 and then produces 2 and then goes on onto infinity, and also that the number sequence is a natural thing, which was created along with everything else in the Beginning when God created the heavens and the earth. Well, no. The number sequence is a human invention and came to be at once and complete when Euclid wrote down its few driving axioms.

Just remember that the number sequence does not depend on time and when its axioms were established, the whole sequence burst forth in all its splendor and utter predictability:

There are many ways to represent the number sequence, but we will never be able to convey it fully in any way. The number sequence is infinite and works only when infinity is observed. There are some beautiful and stunning infinite series that converge upon some finite value. Mind boggling and fascinating, these series illustrate that infinity comes in different sizes, and this glorious phenomenon defies any definition of it. Infinity lives in the number sequence like a mechanical soul, utterly incomprehensible and always fleeting the limitations posed by numbers, and many have noted that math is rather an art form than a science. Maybe it's both and math too gives us another reason to be awed about the things we can not fathom.