Phi (the golden ratio) in prime quadruplets

[Arcw]

Phi exists at the center of prime quadruplets, along with its square root, and cube root!


http://www.code144.com/zphithrice.png


The 'pos' numbers come from the position of the prime numbers in the sequence itself, i.e. 193 is the 44th prime number, and 197 is the 45th prime number, thus, the center of the prime quadruplet (105) gets assigned a value at the center of these (44.5), and these numbers can make Phi!


Do you think it is strange that Phi is hiding here?

 

[HallsofIvy]

I think it is strange that you think phi is "hiding" in there. What you show is the ratio of two half integers, 44.5/27.5= 1.6181818... which is NOT phi. That is a rational number while phi is irrational. Yes, it is "close" to phi (if you consider two decimal places "close") but it is easy to find numbers "close" to any given number if you look long enough.

 

[Arcw]

I think it is strange that you think phi is "hiding" in there. What you show is the ratio of two half integers, 44.5/27.5= 1.6181818... which is NOT phi. That is a rational number while phi is irrational. Yes, it is "close" to phi (if you consider two decimal places "close") but it is easy to find numbers "close" to any given number if you look long enough.

Do you think it is a coincidence that the square root, and cube root (also to approximations) of Phi then directly follow, in that order, and all occur at the center of the quadruplets.

Get a probability guy over here, stat.

And by the way, it's three decimal places, not two -- and the cube root that follows is accurate to 5 decimal places.

 

[Strilanc]

You're not even using the same formula for all of them. Or the same spacing. Or the same column.

What are the odds of finding approximations of three given values when you can manipulate the formula, spacing, and column? Pretty damn good.