Nafat Conjecture (2026): Non-existence of solutions of n^n + R^Z = Z^R over Gaussian primes

We state a conjecture in exponential Diophantine number theory, inspired by Fermat's Last Theorem and Beal's conjecture. The Nafat Conjecture asserts that there is no triplet (n, R, Z), with n a positive integer greater than 2 and R != Z Gaussian primes, such that n^n + R^Z = Z^R, where complex powers are defined with the principal logarithm: a^b = exp(b Log a).

The document gives the definitions, a triviality analysis (the equation has infinitely many solutions over the positive reals, so the Gaussian-prime constraint is essential), and a high-precision numerical search (130 digits) for 3 <= n <= 30 and Gaussian primes R, Z with norm below 1000 (668 Gaussian primes, 12,475,568 triplets tested): no solution was found. The attached Python script (search2.py) reproduces this search.

Status: this is an unproved conjecture, supported by a numerical search with no counterexample in the range above. It is not a proof. The document also lists open questions (modular curves, Mihailescu's theorem, norm study). Bilingual document (French and English).