PERMUTABLE PRIME NUMBERS and CIRCULAR PRIME NUMBERS DECOMPOSITIONS FOR MERSENNES EXPONENTS

We present a novel representation system for integers based on families of permutable primes. We demonstrate that any integer can be expressed in the form: $$N = k \times A + \sum_{i=1}^{n} P_i$$ where $A$ is a specially chosen "anchor" permutable prime from a family with digits $\{9,9,9,3,3,1\}$ (or its higher-dimensional extensions), the $P_i$ are permutable primes drawn from a fixed set of families, and $k$ is a positive integer that typically matches the first digits of $N$. This representation reveals deep connections between Mersenne exponents and permutable prime families, with applications to primality testing and number classification