(I1) odd p
  ok   p=3 q=3: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=3 q=9: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=3 q=27: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=5 q=5: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=5 q=25: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=7 q=7: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=7 q=49: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=11 q=11: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=13 q=13: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
  ok   p=41 q=41: eps*U^(q-1) and eps*(1-U)^(q-1) formulas
(I2) odd p
  ok   p=3 q=3: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=3 q=3: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=3 q=9: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=3 q=9: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=3 q=27: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=3 q=27: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=5 q=5: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=5 q=5: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=5 q=25: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=5 q=25: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=7 q=7: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=7 q=7: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=7 q=49: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=7 q=49: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=11 q=11: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=11 q=11: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=13 q=13: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=13 q=13: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=41 q=41: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=41 q=41: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
  ok   p=43 q=43: deg Pi_q = 4(q-1), Pi_q(0) != 0
  ok   p=43 q=43: deg Omega_q = 4(q+1), Omega_q(0) != 0, a^2 | Omega_q
(I3),(I4) p = 2
  ok   q=2: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=2: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=4: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=4: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=8: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=8: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=16: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=16: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=32: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=32: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=64: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=64: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
  ok   q=128: U^q = U+E_d, eps*U^(q-1) = eps+E_d(1+U), eps*(1+U)^(q-1) = eps+E_d*U
  ok   q=128: deg N_d = 4q, N_d(0)=1, a^2 | N_d, deg N_d/a^2 = 4q-8, (N_d/a^2)(0) = 1
(I5)
  ok   p=2: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=3: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=5: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=7: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=11: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=13: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=17: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=19: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=23: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=29: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=31: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=37: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=41: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=43: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=47: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=53: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=59: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=61: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=67: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=71: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=73: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=79: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=83: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=89: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=97: gcd(1+x^4, x+x^2+x^3) = 1
  ok   p=101: gcd(1+x^4, x+x^2+x^3) = 1
  (1+x+x^2 mod x^4+1 has three non-zero coordinates 1,x,x^2 in the basis 1,x,x^2,x^3; a monomial or binomial
   c1*x^e1 + c2*x^e2 reduces to at most two non-zero coordinates since x^4 = -1: no machine check needed)
(I6) y^n = S_{n-1}(t) y - S_{n-2}(t), t = y + 1/y, over Z
  ok   n <= 40: y^n = S_{n-1} y - S_{n-2} and (y - 1/y) S_{n-1} = y^n - y^-n
ALL IDENTITIES OK
elapsed_seconds 0
