On Generalized Ramanujan-Style Nested Radicals for All Integers

(Internal code: C002840)

Summary / Comment

This work does not originate from classical radical denesting formulas.
Its starting point is an arithmetic observation: in the identity p^2 - q^2= (p-q)(p+q), factor pairs occur at a fixed even distance 2q.

Ramanujan’s famous nested radical for 3 corresponds to the special case q=1.
This immediately extends to square-minus-4, square-minus-9, etc., yielding a parametrized family indexed by q.

From this viewpoint, the paper develops a systematic recursive construction of infinite nested radicals that converge exactly to any prescribed integer x. Denesting identities arise later as a structural consequence, not as an input.

A second, independent realization is obtained via difference-of-oblongs factorizations, showing that the phenomenon reflects a general multiplicative structure rather than a peculiarity of square roots.

The current version corrects and consolidates earlier drafts.