@markgritter @mattmcirvin given the Ulam square spiral and the Gaussian primes:
what's the probability that they're both prime?
What fraction of Ulam spiral primes are also Gaussian primes? What fraction of Gaussian primes are also Ulam spiral primes?
and F[n_] := {Re[#], Im[#]} & /@ Fold[Join[#1, Last[#1] + I^#2 Range[#2/2]] &, {0}, Range[n]] G[n_] := Table[#[[Prime[k]]], {k, 1, PrimePi[n^2/4 + 1]}] &[F[n]] G[20000] ... and their intersection is still {{-1, -1}, {1, -1}, {1, 1}}
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Huh, that's pretty cool. It looks like the norm of all the prime lattice points on the Ulam spiral is even, which is why they're not Gaussian primes. So there's probably a parity argument here.
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The values at the "corners" (+/- x, +/- x) are always odd. But these can't be Gaussian primes because x^2+x^2 is even. Stepping by 2 to another odd number along the spiral (at the same distance) gives the same parity, hence all odd integers have even Gaussian norm?
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