Recursive sufficiency for the Collatz conjecture and computational verification
Year: 2025
Type: article
Abstract: We define the notion of recursive sufficiency for the Collatz conjecture and we use it to present some results concerning the computational verification of the conjecture. For any integer N ≥ 1 and any recursively sufficient set F, it is proved that all integers in the interval [1, N] satisfy the conjecture if and only if F ∩ [1, N] satisfies the conjecture. We offer a sequence of sieves for which the corresponding sequence of elimination percentages tends to 100\%, and as a result, for any integer P arbitrarily close to 100, we give a sieve whose elimination percentage is at least P%. Also, we prove that if N=2(3^n)+1 is the largest known integer for which all integers 1, 2, ..., N satisfy the conjecture, then all integers N+1, N+2, ..., 2N will satisfy the conjecture as well, and hence, they can be eliminated from the verification process.
Author: Mohammad Ansari
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4412928803