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Powered Pair (Posted on 2013-04-12)

Find all possible pairs (a,b) of positive integers such that a^a has b digits and b^b has a digits, where each of a^a and b^b is expressed in base ten.

No Solution Yet Submitted by K Sengupta

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Integers only, please (spoiler)

| Comment 1 of 2

Without loss of generality, assume a <= b

Then b = length(a^a) <= length(b^b) =a

So b <= a

But a <= b, so b = a

We therefore need a^a having a digits

Well, the number of digits in a^a = 1 + [a*log(a)], (where [] is the floor function], so we need a-1<= a*log(a) < a

In other words, (a-1)/a <= log(a) < 1 (so a must be less than 10). A quick check in excel finds that only 1, 8 and 9 work.

Final answer: (1,1), (8,8), (9,9)

Edited on April 12, 2013, 7:13 pm

Posted by Steve Herman on 2013-04-12 14:34:20

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