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 Powered Pair (Posted on 2013-04-12)
Find all possible pairs (a,b) of positive integers such that aa has b digits and bb has a digits, where each of aa and bb is expressed in base ten.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution Comment 2 of 2 |
`list   10   for A=1 to 999   20     Pwr=A^A   30     B=len(cutspc(str(Pwr)))   40     Pwr=B^B   50     if len(cutspc(str(Pwr)))=A then print A;B,A^A,B^B   60   next`
` a  b      a^a            b^b 1  1    1               1 8  8    16777216        16777216 9  9    387420489       387420489Overflow in 50?a,b 183     415OK`

That when we go higher than this, the lengths become larger than is possible for a solution is shown by:

`a     len(a^a)1       12       13       24       35       46       57       68       89       910      1111      1212      1313      1514      1715      1816      2017      2118      2319      2520      2721      2822      3023      3224      3425      3526      3727      3928      4129      4330      4531      4732      4933      51`

So that, for example 33^33's length of 51 would lead to 51^51, which is certainly larger than 33 digits long (in fact, larger than 51 digits long as this series progresses).

 Posted by Charlie on 2013-04-12 16:07:42
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