Classical Constants Generated from Prime Numbers: A Pedagogical Bridge Between the Discrete and the Continuous

We present several constructions of classical constants such as e, π, and the golden ratio φ using only prime numbers or prime–based sequences (for instance, prime indices in the Fibonacci and Lucas sequences). Our goal is not numerical efficiency but to highlight the pedagogical connection between prime arithmetic and the analytic nature of fundamental constants. We show how the Prime Number Theorem and Euler’s product for the Riemann zeta function allow us to reinterpret e and π as “prime-generated” limits or products, and we suggest classroom activities for calculus, real analysis, and introductory number theory courses.