Lemma 3 (Fricke, tr(a_k y) = s(t_k-1), t_{k+1} = f_s(t_k)) verified on 1800 random pairs, p in {5,7,11,13,101,1009}
Lemma 5 verified exhaustively for p in [5, 7, 11, 13, 17]: every (s,t) with s^2 != 4 is (tr y, tr[x,y])
r=1: Q_r=16; least prime p with a point (s!=0): p=7, s=1, h=1, c=3; x=(0, 1, 6, 0), y=(0, 6, 1, 1); Engel tail of (x,y) in PSL(2,7) = 3 > r
r=2: Q_r=484; least prime p with a point (s!=0): p=7, s=1, h=2, c=3; x=(2, 6, 1, 0), y=(0, 6, 1, 1); Engel tail of (x,y) in PSL(2,7) = 4 > r
r=3: Q_r=12544; least prime p with a point (s!=0): p=11, s=3, h=1, c=6; x=(1, 4, 8, 0), y=(0, 10, 1, 3); Engel tail of (x,y) in PSL(2,11) = 5 > r
r=4: Q_r=258064; least prime p with a point (s!=0): p=11, s=4, h=5, c=10; x=(0, 1, 10, 0), y=(0, 10, 1, 4); Engel tail of (x,y) in PSL(2,11) = 6 > r
r=5: Q_r=4682896; least prime p with a point (s!=0): p=13, s=1, h=3, c=3; x=(1, 0, 9, 1), y=(0, 12, 1, 1); Engel tail of (x,y) in PSL(2,13) = 7 > r
r=6: Q_r=79780624; least prime p with a point (s!=0): p=19, s=3, h=1, c=14; x=(2, 1, 18, 0), y=(0, 18, 1, 3); Engel tail of (x,y) in PSL(2,19) = 8 > r
r=7: Q_r=1317109264; least prime p with a point (s!=0): p=19, s=3, h=2, c=14; x=(17, 13, 4, 2), y=(0, 18, 1, 3); Engel tail of (x,y) in PSL(2,19) = 9 > r
r=1: primes 1000<p<3000 with no point (s!=0): [] (count 0)
r=2: primes 1000<p<3000 with no point (s!=0): [] (count 0)
r=3: primes 1000<p<3000 with no point (s!=0): [] (count 0)
r=4: primes 1000<p<3000 with no point (s!=0): [] (count 0)
