A Property of an
It is very easy to show that there exists a power of 3 - 3n - that ends with 001. A quick glance at the proof will reveal that this is also true of powers of other odd numbers (except those divisible by 5.) As a different extension, we may select arbitrary long sequences of zeros that end with a single 1 - for each of them there exists a power of three that ends with this sequence. And, again, the same applies to every odd number excluding multiples of 5.
The differences of powers like (3n - 3m) also have a more generic (though a simpler) property that for any integer A, there exist two powers of 3 whose difference is divisible by A. Both facts are most easily proved with the help of the Pigeonhole Principle.
The latter is also used to prove the following astonishing property:
|(*)||For every sequence of decimal digits S, there exists a power of 3 that starts with the sequence S.|
The same is true for other integers instead of 3. And there is no reason to restrict oneself to the base 10. Any positional system will serve the same purpose.
Say, how many digits of the number π you remember?
We do not know how long the power will be but, regardless of how many digits is takes to write a number, the fact that N starts with a sequence S can be expressed as
S·bk ≤ N < (S + 1)·bk,
where b is the base of the selected system. Replacing N with an and taking logarithms in base b gives
k + logbS ≤ n·logba < k + logb(S+1)
Of course it's not true that there exists a power of 3 that ends with 3141592 in the decimal system.
- R. Honsberger, Ingenuity in Mathematics, MAA, New Math Library, 1970
Copyright © 1996-2018 Alexander Bogomolny