This document describes an RSA encryption method using Pell's equation. It involves: 1) Selecting a secret odd prime integer R and finding the least positive integral solution (Y0, X0) to the Diophantine equation Y^2 - RX^2 = 1. 2) Selecting two large odd primes p and q and defining N = pq. 3) Defining the public key α using Y0, X0, R, and the Euler totient function φ(n). The encryption of a message m involves computing E ≡ mS (mod n) where S is derived from α, and the decryption recovers m by computing E^d (mod n