Exploring the Prime Spiral: Is It a Quirk or an Intriguing Study?

In summary, the prime spiral is a graphical representation of prime numbers on a two-dimensional grid. It is useful for visualizing patterns and relationships among prime numbers and has been used in cryptography and number theory research. There is debate among mathematicians about its significance, but it is closely related to other mathematical concepts and has been used in various fields of research. While it may not have direct practical applications, it serves as a visual representation of the distribution of prime numbers and has implications for understanding complex systems.
  • #1
Helical
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I've was reading about it [http://mathworld.wolfram.com/PrimeSpiral.htm] [Broken] and found it intriguing, has there been a great deal of study devoted to it or is it thought of as some kind of quark?

[P.S. I don't really know much about number theory, just curious]
 
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  • #2
I think it's fairly well-understood. The diagonals are related to prime-generating polynomials (which you can also read about on MathWorld).
 
  • #3
but ALL the diagonals of Ulam spiral generate Polynomials that have prime values or only a few special diagonals of Ulam spiral only generate primes.
 
  • #4
mhill said:
but ALL the diagonals of Ulam spiral generate Polynomials that have prime values or only a few special diagonals of Ulam spiral only generate primes.
Be careful, no polynomial generates only primes indefinitely. I think you meant --only generate primes up to a large value of n --!
 
  • #5
Uh.. sorry , then i meant what are the diagonals that generate primes up to a large value of 'n' or perhaps a bit harder , given a certain prime what is the Polynomial in Ulam Spiral that for a certain integer the Polynomial gives you the prime 'p' are there SERIOUS studies with calculations for Ulam spiral.
 
  • #6
I'm working through that one, give me a bit.Parse tree:

Code:
(Uh.. sorry)
(then i meant
 (
  (
   (
    what are the diagonals
    (that generate primes up to a large value of 'n')
   )
  or perhaps a bit harder ,
   (
    (
     (given a certain prime)
    what is the Polynomial in Ulam Spiral
     (
      (that for a certain integer)
      the Polynomial gives you the prime 'p'
     )
    )
   )
  )
 are there SERIOUS studies with calculations for Ulam spiral.)
)

Semantic re-formation:

Are there serious computational studies addressing:
1. What diagonals generate only primes up to large values of n?
2. Harder: given a prime p, what polynomials P have P(n) = p for some n?

Answer:

There are studies addressing #1. There is a close relationship between prime-generating polynomials and such serious topics as Heegner numbers (about which I know little). UPNT and MathWorld have lots of references.

#2 does not seem difficult. Do you have more conditions, or have I perhaps misunderstood you?
 

1. What is the prime spiral?

The prime spiral is a graphical representation of prime numbers on a two-dimensional grid. It is formed by starting with the number 1 at the center of the grid and spiraling outwards in a counterclockwise direction, with each subsequent number being placed in the next available spot on the grid. Prime numbers, which can only be divided by 1 and themselves, tend to cluster along diagonal lines on the spiral.

2. How is the prime spiral useful?

The prime spiral is useful for visualizing patterns and relationships among prime numbers. It can also be used to identify prime numbers, as they tend to be grouped closely together on the spiral. Additionally, the prime spiral has been used in cryptography and number theory research.

3. Is the prime spiral a quirk or a meaningful study?

This is a subject of debate among mathematicians. Some argue that the clustering of prime numbers on the spiral is just a coincidence and does not hold any deeper significance. Others see the spiral as a fascinating and useful tool for understanding prime numbers and their properties.

4. How is the prime spiral related to other mathematical concepts?

The prime spiral is closely related to the Ulam spiral, which is a similar graphical representation of prime numbers on a grid. It also has connections to mathematical concepts such as number theory, geometry, and graph theory.

5. Are there any real-world applications of the prime spiral?

While the prime spiral may not have direct practical applications, it has been used in various fields of research, such as cryptography and number theory. It also serves as a visual representation of the distribution of prime numbers, which has implications for understanding patterns in nature and other complex systems.

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