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Cycle length

The length of a non-trivial cycle is known to be at least 114 208 327 604 (or 186 265 759 595 without short-cut). If it can be shown that for all positive integers less than 3 × 2 69 {\displaystyle 3\times 2^{69}} the Collatz sequences reach 1, then this bound would raise to 217 976 794 617 (355 504 839 929 without short-cut). In fact, Eliahou (1993) proved that the period p of any non-trivial cycle is of the form p = 301994 a + 17087915 b + 85137581 c {\displaystyle p=301994a+17087915b+85137581c} where a, b and c are non-negative integers, b ≥ 1 and ac = 0. This result is based on the continued fraction expansion of ln 3 / ln 2.