A table from even numbers under the map x ↦→ 2x − 3. Part I. Fermat and Mersenne numbers in the table

We study the two-dimensional table T_{n,k} = f^k(2n), where f(x) = 2x - 3, n ≥ 1, k ≥ 0. The table is generated by applying the same affine map f to every even number 2n, iteratively to the right. We derive a closed form T_{n,k} = 3 + 2^k(2n - 3) and show that after the shift y = x - 3 the map f becomes a pure doubling, so the entire structure of the table is governed by powers of two. Within this table we locate two classical families exactly. The Mersenne numbers M_p = 2^p - 1 occupy the unique position (n, k) = (2^{p-3} + 1, 2) for every p ≥ 3, while the Fermat numbers F_m = 2^{2^m} + 1 occupy (n, k) = (2^{2^m-2} + 1, 1) for every m ≥ 1. Both positions are proved to be unique using the 2-adic valuation. When p = 2^m + 1, the Fermat number F_m and the Mersenne number M_{2^m+1} lie in the same row and satisfy the identity M_{2^m+1} = 2F_m - 3, i.e. one step of f maps Fermat to Mersenne. We introduce the subtable R_j(p) of Mersenne rows, parametrised by a single number M_{p-2}, and prove the closed form R_j(p) = 3 + 2^{j+1} M_{p-2} = 2^{p+j-1} - K_j, where K_j = 2^{j+1} - 3 = -T_{1,j+1} is a correction that is literally written in the first row of the table. Since the first row (n = 1) is the only one that goes to negative values — because its initial distance to the fixed point x = 3 is negative — the sign of the correction is not a convention but a consequence of the geometry of f. We further give binary representations: M_p = 1^p, F_m = 1 0^{2^m - 1} 1, R_j(p) = 1^{p-2} 0^{j-1} 11, K_j = 1^{j-1} 0 1. In binary form the operation x ↦ 2x - 3 "fills zeros with ones", which explains the structural regularity of every column. Finally, we introduce the extended Fermat numbers G_s = 2^s + 1 and formulate the Mersenne–Fermat dichotomy: if Mersenne is treated as the main family, the G_s appear as intermediate terms (including the classical F_m = G_{2^m}); if Fermat is treated as the main family, the Mersenne numbers M_p with p ≠ 2^m + 1 disappear. Both effects have the same numerical measure, 2^m - 1, and both follow from a single condition p - 1 = 2^m. In this sense the dichotomy is a theorem, not a convention. The table T_{n,k} is presented as a new tool for studying the interrelations between classical sequences. To the best of our search, this specific formulation — with the subtable R_j(p), the correction K_j, the binary picture and the Mersenne–Fermat dichotomy — does not occur in the literature available to us; however, we make no claim of absolute novelty, and we explicitly invite readers who know of prior work on the same construction to inform the author. This is Part I of the study; further observations on the table are reserved for subsequent parts.