The relationship between stopping time and number of odd terms in Collatz sequences

2019/11/01 by Ruggiero, Rafael
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.1911.01229

Abstract

The Collatz sequence for a given natural number N is generated by repeatedly applying the map N → 3N+1 if N is odd and N → N/2 if N is even. One elusive open problem in Mathematics is whether all such sequences end in 1 (Collatz conjecture), the alternative being the possibility of cycles or of unbounded sequences. In this paper, we present a formula relating the stopping time and the number of odd terms in a Collatz sequence, obtained numerically and tested for all numbers up to 10⁷ and for random numbers up to 2^128.000. This result is presented as a conjecture, and with the hope that it could be useful for constructing a proof of the Collatz conjecture.

Related

• The intricate labyrinth of Collatz sequences
• Paradoxical behavior in Collatz sequences
• A window to the Convergence of a Collatz Sequence
• On the nonexistence of cycles for the Collatz function
• Observations towards a proof of the Collatz conjecture using 2^jk+x number series