List of the penultimate lucas lehmer step: S[p-3]= + 2^((p+1)/2) or S[p-3]= - 2^((p+1)/2) For the 3 "universal" initial values: S[0]=4, S[0]=10, S[0]=2/3: p=5 or 7 (mod 8) and p>5 <=> penultimate(4)*penultimate(10) = +1 p=1 or 5 (mod 8) and p>5 <=> penultimate(2/3) = +1 S[0]=4 S[0]=10 S[0]=2/3 p%8 M2: p=3 + - - 3 M3: p=5 + - - 5 M4: p=7 - - - 7 M5: p=13 + + + 5 M6: p=17 - + + 1 M7: p=19 - + - 3 M8: p=31 + + - 7 M9: p=61 + + + 5 M10: p=89 - + + 1 M11: p=107 - + - 3 M12: p=127 + + - 7 M13: p=521 - + + 1 M14: p=607 - - - 7 M15: p=1279 - - - 7 M16: p=2203 + - - 3 M17: p=2281 - + + 1 M18: p=3217 - + + 1 M19: p=4253 + + + 5 M20: p=4423 - - - 7 M21: p=9689 - + + 1 M22: p=9941 + + + 5 M23: p=11213 - - + 5 M24: p=19937 + - + 1 M25: p=21701 - - + 5 M26: p=23209 + - + 1 M27: p=44497 - + + 1 M28: p=86243 + - - 3 M29: p=110503 + + - 7 M30: p=132049 + - + 1 M31: p=216091 - + - 3 M32: p=756839 + + - 7 M33: p=859433 - + + 1 M34: p=1257787 - + - 3 M35: p=1398269 + + + 5 M36: p=2976221 + + + 5 M37: p=3021377 + - + 1 M38: p=6972593 + - + 1 M39: p=13466917 + + + 5 M40: p=20996011 + - - 3 M41: p=24036583 - - - 7 M42: p=25964951 + + - 7 M43: p=30402457 - + + 1 M44: p=32582657 - + + 1 M45: p=37156667 + - - 3 M46: p=42643801 - + + 1 M47: p=43112609 + - + 1 M48: p=57885161 - + + 1 M49: p=74207281 - + + 1 M50: p=77232917 + + + 5 M51: p=82589933 - - + 5 25+ 30+ 30+ 25- 20- 20- ----------------------------- 50 50 50 For S0=4 and S0=10 there are 2 infinite sequences of universal initials values, but as can be seen below it seems they are all the same or the opposite of the initial S0=4 and S0=10 penul- timate lists in an alternating + - - + + - - + + pattern. It turns out that the previous steps before the penultimate are not the same. A(1)=4, A(2)=52, A(N)=14*A(N-1)-A(N-2) A(1)=4 A(2)=52 A(3)=724 A(4)=10084 A(5)=140452 A(6)=1956244 A(7)=27246964 A(8)=379501252 A(9)=5285770564 A(10)=73621286644 A(11)=1025412242452 S0=4 A(2) A(3) A(4) A(5) A(6) A(7) A(8) A(9) A(10) A(11) M2: p=3 + - - + + - - + + - - M3: p=5 + - - + + - - + + - - M4: p=7 - + + - - + + - - + + M5: p=13 + - - + + - - + + - - M6: p=17 - + + - - + + - - + + M7: p=19 - + + - - + + - - + + M8: p=31 + - - + + - - + + - - M9: p=61 + - - + + - - + + - - M10: p=89 - + + - - + + - - + + M11: p=107 - + + - - + + - - + + M12: p=127 + - - + + - - + + - - M13: p=521 - + + - - + + - - + + M14: p=607 - + + - - + + - - + + M15: p=1279 - + + - - + + - - + + M16: p=2203 + - - + + - - + + - - M17: p=2281 - + + - - + + - - + + M18: p=3217 - + + - - + + - - + + M19: p=4253 + - - + + - - + + - - M20: p=4423 - + + - - + + - - + + M21: p=9689 - + + - - + + - - + + M22: p=9941 + - - + + - - + + - - M23: p=11213 - + + - - + + - - + + M24: p=19937 + - - + + - - + + - - M25: p=21701 - + + - - + + - - + + M26: p=23209 + - - + + - - + + - - M27: p=44497 - + + - - + + - - + + M28: p=86243 + - - + + - - + + - - M29: p=110503 + - - + + - - + + - - M30: p=132049 + - - + + - - + + - - M31: p=216091 - + + - - + + - - + + M32: p=756839 + - - + + - - + + - - M33: p=859433 - + + - - + + - - + + M34: p=1257787 - + + - - + + - - + + M35: p=1398269 + - - + + - - + + - - B(1)=10, B(2)=970, B(N)=98*B(N-1)-B(N-2) B(1)=10 B(2)=970 B(3)=95050 B(4)=9313930 B(5)=912670090 B(6)=89432354890 B(7)=8763458109130 B(8)=858729462339850 B(9)=84146723851196170 B(10)=8245520207954884810 B(11)=807976833655727515210 S0=10 B(2) B(3) B(4) B(5) B(6) B(7) B(8) B(9) B(10) B(11) M2: p=3 - + + - - + + - - + + M3: p=5 - + + - - + + - - + + M4: p=7 - + + - - + + - - + + M5: p=13 + - - + + - - + + - - M6: p=17 + - - + + - - + + - - M7: p=19 + - - + + - - + + - - M8: p=31 + - - + + - - + + - - M9: p=61 + - - + + - - + + - - M10: p=89 + - - + + - - + + - - M11: p=107 + - - + + - - + + - - M12: p=127 + - - + + - - + + - - M13: p=521 + - - + + - - + + - - M14: p=607 - + + - - + + - - + + M15: p=1279 - + + - - + + - - + + M16: p=2203 - + + - - + + - - + + M17: p=2281 + - - + + - - + + - - M18: p=3217 + - - + + - - + + - - M19: p=4253 + - - + + - - + + - - M20: p=4423 - + + - - + + - - + + M21: p=9689 + - - + + - - + + - - M22: p=9941 + - - + + - - + + - - M23: p=11213 - + + - - + + - - + + M24: p=19937 - + + - - + + - - + + M25: p=21701 - + + - - + + - - + + M26: p=23209 - + + - - + + - - + + M27: p=44497 + - - + + - - + + - - M28: p=86243 - + + - - + + - - + + M29: p=110503 + - - + + - - + + - - M30: p=132049 - + + - - + + - - + + M31: p=216091 + - - + + - - + + - - M32: p=756839 + - - + + - - + + - - M33: p=859433 + - - + + - - + + - - M34: p=1257787 + - - + + - - + + - - M35: p=1398269 + - - + + - - + + - - http://mersenneforum.org/showthread.php?t=5862