Experimental Observations on Cycle Structures of the Mappings 2x-(n+1)⌊x/n⌋ and Possible Connections with Prime Numbers
Year: 2026
Type: preprint
Abstract: In this paper we present new conjectural properties concerning prime numbers, Mersenne numbers and Twin primes. These properties are related to the lengths and the number of cycles generated by the function f_n where n is a positive even integer and for every positive integer x this function is defined as follows : f_n(x) = 2x −(n+1)⌊x/n⌋ Fixed points are excluded from the analysis. An extensive computational exploration of the nontrivial cycles generated by f_n reveals several regularities connecting the cycle structure with arithmetic properties of the parameter (n+1). Based on these observations, we formulate conjectures concerning prime numbers, twin primes, and Mersenne numbers. In particular, The numerical results suggest that, whenever f_n has a unique nontrivial cycle, n+1 is prime and the cycle has length n. On the other hand, when (n+1) is prime, all nontrivial cycles have the same length, which divides n or the function generates a unique cycle of length n. We also observe specific patterns for parameters of the form 2^P, where P is prime, and for 2^k-2, where 2^k-1 is a Mersenne number. These results are purely conjectural and are supported by computational evidence rather than by theoretical proofs. .
Author: Rajab, Raouf
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171464886
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