Assess students' ability to construct and analyze families of nonlinear integer recurrences (e.g., a_{n+1}=a_n^2+c and natural variants). Tasks should include producing explicit examples with chosen c and initial terms, proving rigorous results on asymptotic growth rates (e.g., super-exponential or double-exponential behavior, monotonicity), investigating occurrence of prime terms (finite vs. infinite occurrence, heuristics and proofs where possible), and characterizing divisibility patterns (primitive divisors, periodicity modulo m, and gcd properties). Problems may combine proof-based questions with short computational verifications.