Is it known that the primes on the Ulam prime spiral distribute themselves equally in sectors around the origin? To be specific, say the quadrants? (Each quadrant is closed on one axis and open on the other.) For example, in the $50 \times 50$ spiral below, I count the number of primes in the four quadrants to be $(103,96,88,86)$ ($\sum=373$), leading to ratios $(0.276,0.257,0.236,0.231)$:
     Prime Spiral
For the $500 \times 500$ spiral, I count $(5520,5553,5535,5469)$ ($\sum=22077$) leading to $(0.250,0.252,0.251,0.248)$. Empirically there is a convergence to $\frac{1}{4}$, but I wonder if this has been proven?


  • 2
    $\begingroup$ I have played with those numbers using Ulam's Spiral with these quadrants. The distribution is fairly even. I don't know if anyone has proved anything about it. Here are some rough numbers for $1,712,012,977$ primes. $$\{427997847 \text{nq },428004936 \text{wq },427992540 \text{sq },428017654 \text{eq }\}$$ $$\{0.249996847424597529788 \text{nq },0.250000988164238663945 \text{ wq },0.249993747564916968500 \text{ sq },0.250008416846246837765 \text{ eq }\}$$ $\endgroup$
    – Fred Kline
    Jul 9, 2012 at 0:25
  • $\begingroup$ Now crossposted to MathOverflow. $\endgroup$ Jul 12, 2012 at 21:42
  • 2
    $\begingroup$ And answered there by Terry Tao: Yes, a proof is available via known results that "the square roots $p^{1/2}$ of primes are uniformly distributed modulo 1." $\endgroup$ Jul 13, 2012 at 11:50

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