Super Even

1000 · New Member Offline Joined: 08 Jun 2004 Posts: 7

A number is called super-even if it is a power of 2 and all of its digits are even. Find all super-even numbers.

For example, $2^1=2, 2^2=4, 2^3=8, 2^6=64, 2^{11}=2048$ are all super-even. Note that it may seem easier to prove that there are no more super-even numbers after 2048...but I am looking for a counter-example, which is hard, because you would need a super computer.

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1000 aka. $10^3$

grobber

But $1024$ is not super-even, since it has the digit $1$ , and $2048>1024$ is, so the assertion that there are no more after $1024$ is wrong Confused

Am I missing something here?

**Posted:** Mon Jan 24, 2005 11:28 am

1000 · New Member Offline Joined: 08 Jun 2004 Posts: 7

Oh so sorry, I meant $2^{11}=2048$ , 1024... had a 1 in it.

**Posted:** Mon Jan 24, 2005 11:39 am

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1000 aka. $10^3$

SamE

Well...
There aren't any that have fewer than 1000 digits - I used a calc program to check all the way to 2^3322, when it ran out of memory.
A probability integral tells us there should be 6 1/7, and we've only found five. Interesting.

**Posted:** Tue May 17, 2005 2:57 pm

heartwork

Definitely this is still an open question!
Someone named Bertram Felgenhauer (IMO participant years ago) check it for $2^n$ , $n \leq 2^{69}$ or more. He is still searching. No other values were found and it is supposed there is no other number.
On the other hand there is a claim of proof using some probabilistic reasons. You may see more other facts about at mathforum.org.

**Posted:** Wed May 18, 2005 1:36 am

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