Factorial Building Blocks: The TAN Quest to 1/e

A deep dive into the TAN function—the largest minimum factor in a factorization of n!. We trace the Erdos–Selfridge–Strauss conjecture, the mystique of the lost proof, and Terence Tao’s asymptotic bounds showing tn/n → 1/e. Along the way we unravel how primes just above n/e constrain decompositions and what the convergence rate reveals about the hidden structure of factorials.

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