[zpksvcxm] High powers of 3

A curious pattern of final digits discovered while raising 3 to very high exponents, namely the tower 3^(3^10), 3^(3^100), 3^(3^1000) and so forth. Not only is there a ...0003 tail that extends, but the digits before that seem to converge to something ending in ...459377034485802748997452 .

? for(i=0,9,print(i," ",Mod(3,10^19)^(3^(10^i))))
0 Mod(27, 10000000000000000000)
1 Mod(9572770556941930083, 10000000000000000000)
2 Mod(2822640358094520003, 10000000000000000000)
3 Mod(7420833711745200003, 10000000000000000000)
4 Mod(3130630197452000003, 10000000000000000000)
5 Mod(2815609974520000003, 10000000000000000000)
6 Mod(9086899745200000003, 10000000000000000000)
7 Mod(3948997452000000003, 10000000000000000000)
8 Mod(7489974520000000003, 10000000000000000000)
9 Mod(4899745200000000003, 10000000000000000000)
  *** for: Warning: large exponent in Mod(a,N)^n: reduce n mod phi(N).

We heed the warning, which accelerates computation and allows further investigation. The "lift" function extracts the remainder part of a Mod value. (The .mod member function extracts the modulus.)

? for(i=0,30,print(i," ", Mod(3,10^50)^lift(Mod(3,eulerphi(10^50))^(10^i))))
 0 Mod(27, 10^50)
 1 Mod(49933514477452926276185591240649572770556941930083, 10^50)
 2 Mod(49464382014845383684404622494102822640358094520003, 10^50)
 3 Mod(91879820658144212184935221531007420833711745200003, 10^50)
 4 Mod( 5666529102083590163744828115273130630197452000003, 10^50)
 5 Mod(20514590572323143610626412566862815609974520000003, 10^50)
 6 Mod(95313389858680358587228468921059086899745200000003, 10^50)
 7 Mod(77119826860753014538996853733683948997452000000003, 10^50)
 8 Mod(28419120303269373099024269646147489974520000000003, 10^50)
 9 Mod(22413453720885139735095927392274899745200000000003, 10^50)
10 Mod(45659899372630111116282367002748997452000000000003, 10^50)
11 Mod( 1219314742011767695132978027489974520000000000003, 10^50)
12 Mod(65929886830463330182260580274899745200000000000003, 10^50)
13 Mod(37610648619198624915685802748997452000000000000003, 10^50)
14 Mod(45123797648518558464858027489974520000000000000003, 10^50)
15 Mod(92249122138416515448580274899745200000000000000003, 10^50)
16 Mod( 3605786707258234485802748997452000000000000000003, 10^50)
17 Mod(47514399381890344858027489974520000000000000000003, 10^50)
18 Mod(20797224749703448580274899745200000000000000000003, 10^50)
19 Mod(73295340577034485802748997452000000000000000000003, 10^50)
20 Mod(65262713770344858027489974520000000000000000000003, 10^50)
21 Mod(83557937703448580274899745200000000000000000000003, 10^50)
22 Mod(28659377034485802748997452000000000000000000000003, 10^50)
23 Mod(94593770344858027489974520000000000000000000000003, 10^50)
24 Mod(45937703448580274899745200000000000000000000000003, 10^50)
25 Mod(59377034485802748997452000000000000000000000000003, 10^50)
26 Mod(93770344858027489974520000000000000000000000000003, 10^50)
27 Mod(37703448580274899745200000000000000000000000000003, 10^50)
28 Mod(77034485802748997452000000000000000000000000000003, 10^50)
29 Mod(70344858027489974520000000000000000000000000000003, 10^50)
30 Mod(03448580274899745200000000000000000000000000000003, 10^50)

Wild guess that p-adic numbers have something to do with it.