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Thread: Ulam and Sacks spirals

  1. #1 Ulam and Sacks spirals 
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    Oct 2007
    The Ulam (and later the Sachs) spirals show remarkably regular patterns in the distribution of primes. The famous discovery of Euler that the function f(n) =n^2 - n + 41 gives primes for n = 1...40 can be easily visualised in these spirals.

    In Ulam's spiral you can put any number in the center and still patterns of primes will appear. Apparently these patterns can be generated by playing with b and c in the function f(n) = 4n^2 + bn + c with n positive and counting up 1,2,... Of course these functions don't predict where the primes are, but only descrive small sequences. These regular sequences have been confirmed for very high numbers.

    I'm still intrigued by the Riemann zeros and the fact that they come in symmetrical pairs 1/2 +/- yi. This implies that they can be written as complex zero's of kwadratic functions of the shape: an^2 + bn + c with b/2a = 1/2 and 4ac > b^2.

    I'm aware of the fact that you need to sum an infinite number of zero's to obtain an accurate 'wave' prediction of the number of primes below x, but wondered whether Ulam's kwadratic functions that describe small parts of the distribution of primes, but are at the same time also building blocks of the sum of primes, could have any correlation with the location of the virtual zero's.

    Any hints/tips?

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