# 70 arithmetic/digits/power.two.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 70 arithmetic/digits/power.two.p

Prove that for any 9-digit number (base 10) there is an integral power

of 2 whose first 9 digits are that number.

arithmetic/digits/power.two.s

Let v = log to base 10 of 2.

Then v is irrational.

Let w = log to base 10 of these 9 digits.

Since v is irrational, given epsilon > 0, there exists some natural number

n such that

{w} < {nv} < {w} + epsilon

({x} is the fractional part of x.) Let us pick n for when

epsilon = log 1.00000000000000000000001.

Then 2^n does the job.

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