(2020). Idempotent Factorizations in the Cryptography Classroom. The College Mathematics Journal: Vol. 51, No. 3, pp. 195-203.

RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific $(s,r)$ pairs can be found for which encryption and decryption will be correct for any $(e,d)$ exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime $r\nmid s$ . We give necessary and sufficient conditions for s with this property.