k=  7 m= 3: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k=  9 m= 4: deg phi_(m+1) = 2, (C1) True, (C2) True, (C3) True  -> OK
k= 11 m= 5: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 13 m= 6: deg phi_(m+1) = 3, (C1) True, (C2) True, (C3) True  -> OK
k= 15 m= 7: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 17 m= 8: deg phi_(m+1) = 4, (C1) True, (C2) True, (C3) True  -> OK
k= 19 m= 9: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 21 m=10: deg phi_(m+1) = 5, (C1) True, (C2) True, (C3) True  -> OK
k= 23 m=11: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 25 m=12: deg phi_(m+1) = 6, (C1) True, (C2) True, (C3) True  -> OK
k= 27 m=13: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 29 m=14: deg phi_(m+1) = 7, (C1) True, (C2) True, (C3) True  -> OK
k= 31 m=15: deg phi_(m+1) = 8, (C1) True, (C2) True, (C3) True  -> OK
k= 33 m=16: deg phi_(m+1) = 8, (C1) True, (C2) True, (C3) True  -> OK
k= 35 m=17: deg phi_(m+1) = 9, (C1) True, (C2) True, (C3) True  -> OK
k= 37 m=18: deg phi_(m+1) = 9, (C1) True, (C2) True, (C3) True  -> OK
k= 39 m=19: deg phi_(m+1) = 10, (C1) True, (C2) True, (C3) True  -> OK
k= 41 m=20: deg phi_(m+1) = 10, (C1) True, (C2) True, (C3) True  -> OK
k= 43 m=21: deg phi_(m+1) = 11, (C1) True, (C2) True, (C3) True  -> OK
k= 45 m=22: deg phi_(m+1) = 11, (C1) True, (C2) True, (C3) True  -> OK
k= 47 m=23: deg phi_(m+1) = 12, (C1) True, (C2) True, (C3) True  -> OK
k= 49 m=24: deg phi_(m+1) = 12, (C1) True, (C2) True, (C3) True  -> OK
k= 51 m=25: deg phi_(m+1) = 13, (C1) True, (C2) True, (C3) True  -> OK
k= 53 m=26: deg phi_(m+1) = 13, (C1) True, (C2) True, (C3) True  -> OK
k= 55 m=27: deg phi_(m+1) = 14, (C1) True, (C2) True, (C3) True  -> OK
k= 57 m=28: deg phi_(m+1) = 14, (C1) True, (C2) True, (C3) True  -> OK
k= 59 m=29: deg phi_(m+1) = 15, (C1) True, (C2) True, (C3) True  -> OK
k= 61 m=30: deg phi_(m+1) = 15, (C1) True, (C2) True, (C3) True  -> OK
k= 63 m=31: deg phi_(m+1) = 16, (C1) True, (C2) True, (C3) True  -> OK
k= 65 m=32: deg phi_(m+1) = 16, (C1) True, (C2) True, (C3) True  -> OK
k= 67 m=33: deg phi_(m+1) = 17, (C1) True, (C2) True, (C3) True  -> OK
k= 69 m=34: deg phi_(m+1) = 17, (C1) True, (C2) True, (C3) True  -> OK
k= 71 m=35: deg phi_(m+1) = 18, (C1) True, (C2) True, (C3) True  -> OK
k= 73 m=36: deg phi_(m+1) = 18, (C1) True, (C2) True, (C3) True  -> OK
k= 75 m=37: deg phi_(m+1) = 19, (C1) True, (C2) True, (C3) True  -> OK
k= 77 m=38: deg phi_(m+1) = 19, (C1) True, (C2) True, (C3) True  -> OK
k= 79 m=39: deg phi_(m+1) = 20, (C1) True, (C2) True, (C3) True  -> OK
k= 81 m=40: deg phi_(m+1) = 20, (C1) True, (C2) True, (C3) True  -> OK
k= 83 m=41: deg phi_(m+1) = 21, (C1) True, (C2) True, (C3) True  -> OK
k= 85 m=42: deg phi_(m+1) = 21, (C1) True, (C2) True, (C3) True  -> OK
k= 87 m=43: deg phi_(m+1) = 22, (C1) True, (C2) True, (C3) True  -> OK
k= 89 m=44: deg phi_(m+1) = 22, (C1) True, (C2) True, (C3) True  -> OK
k= 91 m=45: deg phi_(m+1) = 23, (C1) True, (C2) True, (C3) True  -> OK
k= 93 m=46: deg phi_(m+1) = 23, (C1) True, (C2) True, (C3) True  -> OK
k= 95 m=47: deg phi_(m+1) = 24, (C1) True, (C2) True, (C3) True  -> OK
k= 97 m=48: deg phi_(m+1) = 24, (C1) True, (C2) True, (C3) True  -> OK
k= 99 m=49: deg phi_(m+1) = 25, (C1) True, (C2) True, (C3) True  -> OK
k=101 m=50: deg phi_(m+1) = 25, (C1) True, (C2) True, (C3) True  -> OK
k=103 m=51: deg phi_(m+1) = 26, (C1) True, (C2) True, (C3) True  -> OK
k=105 m=52: deg phi_(m+1) = 26, (C1) True, (C2) True, (C3) True  -> OK
k=107 m=53: deg phi_(m+1) = 27, (C1) True, (C2) True, (C3) True  -> OK
k=109 m=54: deg phi_(m+1) = 27, (C1) True, (C2) True, (C3) True  -> OK
k=111 m=55: deg phi_(m+1) = 28, (C1) True, (C2) True, (C3) True  -> OK
k=113 m=56: deg phi_(m+1) = 28, (C1) True, (C2) True, (C3) True  -> OK
k=115 m=57: deg phi_(m+1) = 29, (C1) True, (C2) True, (C3) True  -> OK
k=117 m=58: deg phi_(m+1) = 29, (C1) True, (C2) True, (C3) True  -> OK
k=119 m=59: deg phi_(m+1) = 30, (C1) True, (C2) True, (C3) True  -> OK
k=121 m=60: deg phi_(m+1) = 30, (C1) True, (C2) True, (C3) True  -> OK
k=123 m=61: deg phi_(m+1) = 31, (C1) True, (C2) True, (C3) True  -> OK
k=125 m=62: deg phi_(m+1) = 31, (C1) True, (C2) True, (C3) True  -> OK
k=127 m=63: deg phi_(m+1) = 32, (C1) True, (C2) True, (C3) True  -> OK
k=129 m=64: deg phi_(m+1) = 32, (C1) True, (C2) True, (C3) True  -> OK
k=131 m=65: deg phi_(m+1) = 33, (C1) True, (C2) True, (C3) True  -> OK
k=133 m=66: deg phi_(m+1) = 33, (C1) True, (C2) True, (C3) True  -> OK
k=135 m=67: deg phi_(m+1) = 34, (C1) True, (C2) True, (C3) True  -> OK
k=137 m=68: deg phi_(m+1) = 34, (C1) True, (C2) True, (C3) True  -> OK
k=139 m=69: deg phi_(m+1) = 35, (C1) True, (C2) True, (C3) True  -> OK
k=141 m=70: deg phi_(m+1) = 35, (C1) True, (C2) True, (C3) True  -> OK
k=143 m=71: deg phi_(m+1) = 36, (C1) True, (C2) True, (C3) True  -> OK
k=145 m=72: deg phi_(m+1) = 36, (C1) True, (C2) True, (C3) True  -> OK
k=147 m=73: deg phi_(m+1) = 37, (C1) True, (C2) True, (C3) True  -> OK
k=149 m=74: deg phi_(m+1) = 37, (C1) True, (C2) True, (C3) True  -> OK
k=151 m=75: deg phi_(m+1) = 38, (C1) True, (C2) True, (C3) True  -> OK
k=153 m=76: deg phi_(m+1) = 38, (C1) True, (C2) True, (C3) True  -> OK
k=155 m=77: deg phi_(m+1) = 39, (C1) True, (C2) True, (C3) True  -> OK
k=157 m=78: deg phi_(m+1) = 39, (C1) True, (C2) True, (C3) True  -> OK
k=159 m=79: deg phi_(m+1) = 40, (C1) True, (C2) True, (C3) True  -> OK
k=161 m=80: deg phi_(m+1) = 40, (C1) True, (C2) True, (C3) True  -> OK
k=163 m=81: deg phi_(m+1) = 41, (C1) True, (C2) True, (C3) True  -> OK
k=165 m=82: deg phi_(m+1) = 41, (C1) True, (C2) True, (C3) True  -> OK
k=167 m=83: deg phi_(m+1) = 42, (C1) True, (C2) True, (C3) True  -> OK
k=169 m=84: deg phi_(m+1) = 42, (C1) True, (C2) True, (C3) True  -> OK
k=171 m=85: deg phi_(m+1) = 43, (C1) True, (C2) True, (C3) True  -> OK
k=173 m=86: deg phi_(m+1) = 43, (C1) True, (C2) True, (C3) True  -> OK
k=175 m=87: deg phi_(m+1) = 44, (C1) True, (C2) True, (C3) True  -> OK
k=177 m=88: deg phi_(m+1) = 44, (C1) True, (C2) True, (C3) True  -> OK
k=179 m=89: deg phi_(m+1) = 45, (C1) True, (C2) True, (C3) True  -> OK
k=181 m=90: deg phi_(m+1) = 45, (C1) True, (C2) True, (C3) True  -> OK
k=183 m=91: deg phi_(m+1) = 46, (C1) True, (C2) True, (C3) True  -> OK
k=185 m=92: deg phi_(m+1) = 46, (C1) True, (C2) True, (C3) True  -> OK
k=187 m=93: deg phi_(m+1) = 47, (C1) True, (C2) True, (C3) True  -> OK
k=189 m=94: deg phi_(m+1) = 47, (C1) True, (C2) True, (C3) True  -> OK
k=191 m=95: deg phi_(m+1) = 48, (C1) True, (C2) True, (C3) True  -> OK
k=193 m=96: deg phi_(m+1) = 48, (C1) True, (C2) True, (C3) True  -> OK
k=195 m=97: deg phi_(m+1) = 49, (C1) True, (C2) True, (C3) True  -> OK
k=197 m=98: deg phi_(m+1) = 49, (C1) True, (C2) True, (C3) True  -> OK
k=199 m=99: deg phi_(m+1) = 50, (C1) True, (C2) True, (C3) True  -> OK
k=201 m=100: deg phi_(m+1) = 50, (C1) True, (C2) True, (C3) True  -> OK
PART 1: all conditions hold for odd k in [7, 201]
PART 2 (exact expansion for k = 7, 9):
  k=7: coefficient of t^1 in p q^6 P (minus k beta p (q^6 - 1)) reduces to 0
  k=7: coefficient of t^4 in p q^6 P  reduces to 0
  k=7: coefficient of t^7 in p q^6 P  reduces to 0
  k=7: coefficient of t^10 in p q^6 P  reduces to 0
  k=7: coefficient of t^13 in p q^6 P  reduces to 0
  k=7: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 1.59e-56; |sigma| = 13.0499
  k=7: EXACT IDENTITY VERIFIED
  k=9: coefficient of t^1 in p q^8 P (minus k beta p (q^8 - 1)) reduces to 0
  k=9: coefficient of t^5 in p q^8 P  reduces to 0
  k=9: coefficient of t^9 in p q^8 P  reduces to 0
  k=9: coefficient of t^13 in p q^8 P  reduces to 0
  k=9: coefficient of t^17 in p q^8 P  reduces to 0
  k=9: numeric check of t = (1/(m sigma)) sum lam^-j P(lam^j t) at three points: max error 2.26e-57; |sigma| = 1318.16
  k=9: EXACT IDENTITY VERIFIED
RESULT: PASS
