powers starting with 1111...1 in decimal

rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

is it possible for
a) square
b) cube
of natural number (in decimal system) to start with arbitrarily long sequence of 1's ?

**Posted:** Sun Oct 18, 2009 10:39 am

t0rajir0u

Why not? Round the square root of $1111...$ .

**Posted:** Sun Oct 18, 2009 11:26 am

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PhilG · Poincare Conjecture Offline Joined: 24 Aug 2009 Posts: 173

For the case of squares a number starting with $n$ threes will start with $n - 1$ ones.
For cubes you can use the cube of $\lfloor{10^n \frac {\sqrt [3]{3}}{3}}\rfloor$

**Posted:** Mon Oct 19, 2009 11:52 am

rex88 · P versus NP Offline Joined: 12 Sep 2009 Posts: 22

thanks very much.

what about generalisation (higher powers) ?

**Posted:** Mon Oct 19, 2009 12:38 pm

t0rajir0u

Exactly the same strategy works.

**Posted:** Mon Oct 19, 2009 2:24 pm

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