Mersenne Exponents Structured Calculus - novel framework A simple decimal reverse counting system with zero-insertion rules
Martin Doina
"We define a reverse-decimal evolution on digit strings. Starting from a one-digit seed written with a leading zero, digits evolve independently according to a fixed reverse cycle. At selected evolution levels, a leading zero is inserted, creating a new level of the process. Each initial seed generates a distinct family of values. Empirically, the union of these families coincides with the 52 known Mersenne prime exponents."
We start: 00, 01, 02, 03, 04, 05, 06, 07, 08, 09.
The cycle: `1 → 0 → 9 → 8 → 7 → 6 → 5 → 4 → 3 → 2 → 1`
We apply the operation to each digit counting backward so each column repeat the cicle.
Example:
02
91
80
79
68
57
46
35
24
13
"This provides an alternative structural framework for studying integers."
"Within this framework, the known Mersenne exponents occupy reproducible positions that merit further investigation."
Factorization describes the multiplicative structure of integers. My research investigates whether integers also possess a deterministic structural organization under decimal evolution
Can numbers be organized by a deterministic decimal evolution in a way that reveals meaningful structure, with the Mersenne exponents serving as one well-known reference set?
https://zenodo.org/records/21736221
https://github.com/gatanegro/MERSENNE-COLLATZ/tree/main/Structured%20Calculus
What happens if we apply this framework to all integers sequences?
In my repo github there are light HTML apps to reproduce the calculus structured forward and backward for more study.
Thanks foryour time and if you are interested I will appreciate your help.
Start Nr.
Mersennes Exponents with same start
01
89, 4423, 9689, 21701, 37156667, 42643801,
02
13, 1279, 11213, 216091, 30402457, 32582657
03
2203, 2281, 110503, 1398269, 74207281
04
19937, 6972593
05
61, 127, 132049, 24036583, 57885161,
06
17, 3217, 756839, 13466917, 25964951, 77232917
07
107, 607, 9941, 136279841
08
19, 31, 4253, 44497
09
521, 23209, 86243, 1257787, 2976221, 43112609
00
859433, 3021377, 20996011, 82589933
Structured calculus:
1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1