@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?
I just computed p3 = Select[ Flatten[Table[x + I y, {x, -10000, 10000}, {y, -10000, 10000}]], PrimeQ[#, GaussianIntegers -> True] &] Table[{Re[p3[[n]]], Im[p3[[n]]]}, {n, 1, Length[p3]}]
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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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