The arithmetic-geometric mean (AGM) sequence, classically defined over the real and complex numbers, can be transplanted to a finite field F_q; this concept develops that construction for prime powers q congruent to 5 modulo 8 using only elementary number theory rather than elliptic curves. Each sequence is encoded as a directed 'jellyfish' graph whose vertices are admissible pairs and whose edges follow the arithmetic-mean/geometric-mean recurrence, and one studies its structure: how many parents and children a vertex has, why every nontrivial connected component must contain a cycle, and when such components are guaranteed to exist.
This paper extends the arithmetic-geometric mean (AGM) sequence—classically the pairwise iteration replacing a pair (a,b) by its arithmetic mean and geometric mean, studied over the real and complex …