Lothar 2.20

Born on July 6, 1910, in Amsberg, Westphalia, Lothar Collatz made a significant contribution to the field of mathematics at a relatively young age. During his post-doctoral studies, at just 27 years old, he formulated what has become known as the Collatz Conjecture—a problem that continues to intrigue mathematicians today.

The Collatz Conjecture can be succinctly described as follows:

• Define the function f(n):
  • f(n) = n ÷ 2, if n is even
  • f(n) = 3×n + 1, if n is odd
• For any positive integer n, repeated application of this function appears to always lead to the cycle {4, 2, 1}. In other words, regardless of the starting number, the sequence will eventually decrease and settle into this recurring loop. Although extensive computational evidence supports this behavior, a formal proof remains elusive, and no other cycles have been identified in the sequences.

The concept of the reverse function, denoted as f', provides a means of tracing backward through the sequence. It is defined as follows:

• f'(n):
  • n × 2, if n ≡ 0, 1, 2, 3, or 5 modulo 6
  • (n - 1) ÷ 3, if n ≡ 4 modulo 6 and the "Previous" function is activated
  • n × 2, if n ≡ 4 modulo 6 and the "Multiply" option is selected

This implementation was developed by Jeffrey C. Jacobs and is protected under Copyright ©2010 by TimeHorse, LLC. The source code is available upon request for those interested in further exploration.