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Finding the seventh root of 19203908986159

This Math SE question seems destined for deletion, so before that happens I'm repatriating my answer here. The question is simply:

Let !!n!! be an integer, and you are given that !!n^7=19203908986159!!. How would you solve for !!n!! without using a calculator?

If you haven't seen this sort of thing before, you may not realize that it's easier than it looks because !!n!! will have only two digits.

The number !!n^7!! is !!14!! digits long so its seventh root is !!\left\lceil \frac {14}7\right\rceil=2!! digits long. Let's say the digits of !!n!! are !!p!! and !!q!! so that the number we seek is !!n=10p+q!!.

The units digit of !!n^7!! is odd so !!q!! is odd. Clearly !!q\ne 1!! and !!q\ne 5!!. Units digits of numbers ending in !!3,7,9!! repeat in patterns:

$$ \begin{array}{rl} 3 & 3, 9, 7, 1, 3, 9, \mathbf 7, \ldots \\ 7 & 7, 9, 3, 1, 7, 9, \mathbf 3, \ldots \\ 9 & 9, 1, 9, 1, 9, 1, \mathbf 9, \ldots \end{array} $$

so that !!(10p+q)^7!! can end in !!9!! only if !!q=9!!. Let's rewrite !!10p+9!! as !!10(p+1)-1!!.

Expanding !!(10(p+1)-1)^7!! with the binomial theorem, we get

$$10^7(p+1)^7 - 7\cdot10^6(p+1)^6 + \binom72 10^5(p+1)^5 - \cdots$$

The first term here is by far the largest; it is more than !!\frac{10}7p!! times as large as the second largest. Ignoring all but the first term, we want $$(p+1)^7 \approx 1920390.$$

!!2^7!! is only !!128!!, far too small. !!5^7!! is only !!78125!!, also too small. But !!8^7 = 2^{21} \approx 2000000!! because !!2^{10}\approx 1000!!. This is just right, so !!p+1=8!! and the final answer is $$79.$$

If the !!n!! were a three digit number, these kinds of tricks wouldn't be sufficient, and I would use more systematic and general methods. But I think it's a nice example of how far you can get with mere tricks and barely any theory.

This post was brought to you by the P.D.Q. Bernoulli Intitute of Lower Mathematics. It is dedicated to Gian Pietro Farina. I told him I planned to blog more, and he said he was especially looking forward to my posts about math. And then I posted like eleven articles in a row about Korean dog breed names and goose snot.

[ Addendum 20230219: Roger Crew pointed out that I forgot the binomial coefficients in the binomial theorem expansion. I have corrected this. ]

[ Addendum 20230228: Roger Crew also pointed out a possibly simpler way to find !!p!!. ]

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