The benchmark problem
Given a modulus N, a starting value x, and a step count T, predict the residue after squaring modulo N exactly T times.
x₀2² mod 774² mod 7716² mod 7725² mod 77The answer is 9. Each arrow is one serial step: the next value depends on the residue produced by the previous square.
Prompts use field markers followed by decimal digits. The example above is represented conceptually as:
N is part of every prompt. Its secret prime factors p and q are never supplied to the submission. The target is the decimal representation of y.
An example counts as correct only when every target token is correct. A nearly right residue earns no partial credit for that example.
Hard ranks by the largest consecutively certified T on modulus identities seen during training, then by the corresponding Max T on unseen interpolation and extrapolation modulus sizes. Ties use accuracy at each profile's first uncertified rung. Every example in every rung through Max T must be exactly correct.
Hard is a hidden task evaluation and may change aspects of the recurrence itself; do not assume it is repeated squaring.
Without the factorization of N, the best known general method carries out the T modular squarings in order. Later states depend on earlier ones.
The evaluator generates N = pq and keeps p and q private. It can reduce the exponent with φ(N) and compute exact answers efficiently.
Larger T asks for a longer computation. This makes the task a clean test of recurrent depth, adaptive computation, and extrapolation beyond training depths.
φ(N) = (p − 1)(q − 1)e = 2T mod φ(N)y = xe mod NThe design question