n=11; précision de 11 décimales pour avoir les 10 bonnes premières.
134217728=Mod[8^9,10^n]
16763596801 = Mod[7^134217728,10^n]
Conversion en base 2
IntegerDigits[16763596801,2]
{1,1,1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1}
16763596801 - 2^33-2^32-2^31-2^30-2^29-2^26-2^25-2^24-2^21-2^20-2^13-2^12-2^0 =0
sa13=Mod[Mod[6^2^12,10^11]*Mod[6^2^13,10^11]*6,10^11]=94268676096
sa24=Mod[sa13*Mod[6^2^20,10^11]*Mod[6^2^21,10^11]*Mod[6^2^24,10^11],10^11]=8460527616
sa26=Mod[sa24*Mod[6^2^25,10^11]*Mod[6^2^26,10^11],10^11]=2987063296
sa29=Mod[sa26*Mod[6^2^29,10^11],10^11]=86334611456
a30=Mod[Mod[6^2^29,10^11]*Mod[6^2^29,10^11],10^11]=13491920896
a31=Mod[a30*a30,10^11]=63921442816
a32=Mod[a31*a31,10^11]= 79158009856
a33=Mod[a32*a32,10^11]=62593140736
result=Mod[Mod[sa29*a30,10^11]*Mod[a31*a32,10^11]*a33,10^11]=54204000256
IntegerDigits[54204000256,2]
{1,1,0,0,1,0,0,1,1,1,1,0,1,1,0,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0}
54204000256-2^35-2^34-2^31-2^28-2^27-2^26-2^25-2^23-2^22-2^19-2^18-2^17-2^16-2^14-2^13-2^11=0
sb28=Mod[Mod[Mod[5^2^28,10^11]*Mod[5^2^27,10^11]*Mod[5^2^26,10^11]*
Mod[5^2^25,10^11],10^11]*Mod[Mod[5^2^23,10^11]*
Mod[5^2^22,10^11]*Mod[5^2^19,10^11]*Mod[5^2^18,10^11],10^11]*
Mod[Mod[5^2^17,10^11]*Mod[5^2^17,10^11]*Mod[5^2^16,10^11]*
Mod[5^2^14,10^11]*Mod[5^2^13,10^11]*Mod[5^2^13,10^11],10^11],10^11] = 18212890625
b29=Mod[5^2^29,10^11]=18212890625
b30=Mod[b29*b29,10^11]=18212890625
b31=Mod[b30*b30,10^11]=18212890625
b32=Mod[b31*b31,10^11]=18212890625
b33=Mod[b32*b32,10^11]=18212890625
b34=Mod[b33*b33,10^11]=18212890625
b35=Mod[b34*b34,10^11]=18212890625
sb35=Mod[sb28*b31*b34*b35,10^11]=18212890625
IntegerDigits[18212890625,2]
{1,0,0,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,1}
18212890625-2^34-2^29-2^28-2^27-2^26-2^24-2^23-2^20-2^17-2^15-2^13-2^11-2^0= 0
sc29=Mod[Mod[Mod[4^2^29,10^11]*Mod[4^2^28,10^11]*Mod[4^2^27,10^11]*Mod[4^2^26,10^11]*
Mod[4^2^24,10^11],10^11]*Mod[Mod[4^2^23,10^11]*Mod[4^2^20,10^11]*Mod[4^2^17,10^11]*
Mod[4^2^15,10^11]*Mod[4^2^13,10^11]*Mod[4^2^11,10^11]*4,10^11],10^11]=85560776704
c30=Mod[Mod[4^2^29,10^11]*Mod[4^2^29,10^11],10^11]=55944646656
c31=Mod[c30*c30,10^11]=64691982336
c32=Mod[c31*c31,10^11]=61336016896
c33=Mod[c32*c32,10^11]=66397474816
c34=Mod[c33*c33,10^11]=41354233856
tot34=Mod[sc29*c34,10^11]=18212890624
IntegerDigits[18212890624,2]
{1,0,0,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0}
18212890624-2^34-2^29-2^28-2^27-2^26-2^24-2^23-2^20-2^17-2^15-2^13-2^11=0
stot29=Mod[Mod[Mod[3^2^29,10^11]*Mod[3^2^28,10^11]*Mod[3^2^27,10^11]*
Mod[3^2^26,10^11]*Mod[3^2^24,10^11],10^11]*Mod[Mod[3^2^23,10^11]*
Mod[3^2^20,10^11]*Mod[3^2^17,10^11]*Mod[3^2^15,10^11]*
Mod[3^2^13,10^11]*Mod[3^2^11,10^11],10^11],10^11]=75161036801
Mod[3^2^29,10^11]=71954749441
Mod[71954749441*71954749441,10^11]=17089812481
Mod[17089812481*17089812481,10^11]=35743375361
Mod[35743375361*35743375361,10^11]=97341880321
Mod[97341880321*97341880321,10^11]=27887063041
Mod[27887063041*27887063041,10^11]=52708167681
Mod[52708167681*75161036801,10^11]=84919828481
IntegerDigits[84919828481,2]
{1,0,0,1,1,1,1,0,0,0,1,0,1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
84919828481-2^36-2^33-2^32-2^31-2^30-2^26-2^24-2^23-2^20-2^19-2^18-2^16-2^15-2^14-2^0=0
Mod[Mod[Mod[2^2^30,10^11]*Mod[2^2^26,10^11]*Mod[2^2^24,10^11]*Mod[2^2^23, 10^11],10^11]*
Mod[Mod[2^2^20,10^11]*Mod[2^2^19,10^11]*Mod[2^2^18,10^11]*
Mod[2^2^16,10^11]*Mod[2^2^15,10^11]*Mod[2^2^14,10^11]*2,10^11],10^11]= 68722624512
puis31= Mod[Mod[2^2^30,10^11]*Mod[2^2^30,10^11],10^11]=55944646656
puis32=Mod[55944646656*55944646656,10^11]=64691982336
puis33=Mod[64691982336*64691982336,10^11]=61336016896
puis34=Mod[61336016896*61336016896,10^11]=66397474816
puis35=Mod[puis34*puis34,10^11]=41354233856
puis36=Mod[puis35*puis35,10^11]=16736628736
result=Mod[68722624512*puis36*puis33*puis32*puis31,10^11]=88.170.340.352