Visualisation of prime numbers
The Ulam spiral or prime spiral (discovered by the mathematician Stanislaw Ulam in 1963) is a simple method of visualising the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes.
Ulam created this spiral by first writing positive integers in a spiral as shown below:
He then circled only the prime numbers:
As seen above the odd prime numbers(i.e. all prime numbers except 2) tended to link in diagonal lines.
It has since been proven that this holds true even for large numbers.
Above is an ulam spiral from the number 1 to 160,000, where each number is a pixel.
Tests so far confirm that there are diagonal lines even when many numbers are plotted. The pattern also seems to appear even if the number at the center is not 1 (and can, in fact, be much larger than 1). This implies that there are many integer constants b and c such that the function:
Personal note: I do not know the use of this. If anyone does, feel free to let me know. But I just thought it was kind of cool.
If you want to read more about this, I found a pretty cool site where this guy goes into this subject with a more computerised approach: http://ulamspiral.com/generatePage.asp?ID=1