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    [OC] Prime Distribution in the Sacks Spiral - 60,000 Integers, Euler's Polynomial Highlighted
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    [OC] Prime Distribution in the Sacks Spiral – 60,000 Integers, Euler’s Polynomial Highlighted


    Source: CalculateQuick (visualization), Robert Sacks (1994/2003), Euler's prime-generating polynomial (1772). Prime density reference: Zagier, "The first 50 million prime numbers," Mathematical Intelligencer Vol. 1, 1977.

    Tools: Python with NumPy for sieve computation and Matplotlib for polar rendering. Archimedean spiral coordinates r = √n, θ = 2π√n. 60,000 integers plotted; primality via Sieve of Eratosthenes (validated against trial division for full range).

    The orange curve traces Euler's polynomial f(k) = k² + k + 41, which famously produces primes for every integer k from 0 to 39 – and maintains a 74.7% prime rate across the 245 values within this range. First composite value occurs at k = 40, yielding 1681 = 41².

    Posted by CalculateQuick

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    4 Comments

    3. cryptotope on

      I really appreciate the horizontal gap from the center to the right side, and I enjoyed thinking about why it was there.

      Since θ = 2π√*n*, there’s one full rotation of the spiral – an angle of 2Ï€ radians – every time √*n* is an integer: in other words, every time *n* is a perfect square. (Let’s say *n* = *k*^(2). It cannot be prime, because its factors are 1, *k*, and *n*.)

      But the gap is *two* rows wide, which had me stop and think about it. Why are all the values on one side also all non-prime?

      >!Immediately beside *n* is *n* – 1, which equals *k**^(2)* – 1. That expression can be factored as (*k* + 1)(*k* – 1): two integer factors.!<