The authors conjecture that an odd non-square N is prime iff a quadratic non-residue q, with q not congruent to -1, passes both the Euler criterion and a quadratic-extension binomial congruence.
Euler’s criterion,
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PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts
The authors conjecture that an odd non-square N is prime iff a quadratic non-residue q, with q not congruent to -1, passes both the Euler criterion and a quadratic-extension binomial congruence.