Calculating Tetration

[andydude]

I felt like calculating the last few digits of lots of simple tetrations, because now that I can do it so easily, I feel like it is my duty.

I checked the \( 4 \times 5^d \) period for other bases, and for some reason it worked perfectly, I have no idea why! For example, anyone with an arbitrary-arithmetic calculator (like UNIX bc, just type "echo 4^4^4 | bc") can find that 4^^3 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 so I figured I would start with this. The code I used to calculate the last few digits is very simple. In Mathematica is looks like this:

Code:

PowerDigit[base_, exp_, d_] :=
  With[
    {ls = IntegerDigits[base^exp, 10]},
    If[Length[ls] < d, 0, ls[[-d]]]
  ]

PowerLastFew[base_, exp_, places_] :=
  (* Create a list *)
  Table[
  
    (* Calculate an equivalent digit *)
    PowerDigit[base,
    
      (* using the 4 * 5^d formula in the previous post *)
      Mod[exp - 1, 4 * 5^d] + 1,
      
      (* this is because of 1-based lists *)
      d + 1],
      
    (* where the index 'd' goes from 'places' to 0 *)
    {d, places, 0, -1}
  ]

Using this with base-4 now, and the last few digits I got were ...49006084096 so I'm thinking that the formula works. Going on to calculate some other tetrations:

• 2^^4 = 65536
  
• 2^^5 = ...5719156736 (by naive method: same)
  
• 2^^6 = ...7437428736 (not previously calculable)!
  

• 3^^2 = 27
  
• 3^^3 = 7625597484987 (by naive method: same)
  
• 3^^4 = ...30214289387 (not previously calculable)!
  

• 4^^2 = 256
  
• 4^^3 = ...49006084096 (by naive method: same)
  
• 4^^4 = ...95261392896 (not previously calculable)!
  

• 5^^2 = 3125
  
• 5^^3 = ...80908253125? (by naive method: ...80908203125)?
  
• 5^^4 = ...36371253125? (not previously calculable)?
  

However, as you can see from base-5, one of my assumptions is wrong. Something to deal with another day.

Andrew Robbins