3 ms·
This is probably related to Prime Generating Polynomials [1]. Euler first noticed that `n^2 + n + 41` is prime for the 40 integers n = 0, 1, 2, ..., 39. A secon
by thomasahle 8y ago
This is probably related to Prime Generating Polynomials [1]. Euler first noticed that `n^2 + n + 41` is prime for the 40 integers n = 0, 1, 2, ..., 39. A second-order polynomial should look more or less like a line when plotted around a spiral like this.
There are also third-order prime polynomials, which you might be able to see when visualizing the primes in 3d.
[1]: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
- xyzzyz 8y agoYes, that's right. The real question is, why some quadratic polynomials are so rich with primes? Nobody really has any idea.
- mgsouth 8y agoIt's simpler than that. All primes greater than 2 are equal to 2n + 1 for some n. And since each side of a spiral is 2 longer than the adjacent one, you get a slanted line. You'd have just a blob of bunches of line, except that they thin out. (Primes > 6 are 2 * 3 * n +/- 1, then 2 * 3 * 5 * n +/- [1, 7, 11, 13]. Also, prime series are palindromic. Integers relatively prime to (2, 3, ... p[m]) form a repeating series with period 2 * 3 * ... * p[m], and within each series they are the integers not "knocked out" by the given primes; e.g., 2 * 3 * ... * p[m] +/- [p[m+1], p[m+2], ...]. So the pattern is symmetrical over the period.
- deleted 8y ago[deleted]