1. Principle
Only the residue class \(n\equiv 3\pmod{4}\) requires non-trivial simulation. Even numbers and numbers congruent to 1 modulo 4 descend in a fixed small number of steps.
2. Verified Result
Theorem (computational)
Every integer \(n\) with \(2\le n<2^{36}\) satisfies \(T^{s}(n)<n\) for some positive integer \(s\). Consequently the Collatz conjecture holds for all positive integers less than \(2^{36}\).
The verification completed in 157 seconds using exact integer arithmetic.
3. Implementation
Usage:
gcc -O2 -o verify verify.c && ./verify 36