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86 at most (Posted on 2012-08-18) Difficulty: 3 of 5
286 is conjectured to be the largest power of 2 not containing a zero.
This simply stated conjecture has proven itself to be proof-resistant.

Source: several articles on the web
Do not try to find a counterexample.
Try to explain why it sounds true and why it is so hard to prove.

See The Solution Submitted by Ady TZIDON    
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simple version | Comment 1 of 2

1. Start with the idea that this has already been checked up to 2^n, where n=4.6*10^7. (see http://mathworld.wolfram.com/Zero.html )

2. Based on something like 'old-fogey logic' in 'Start as you wish', assume the digits of 2^n are randomly distributed from 0-9. Actually the distribution don't even need to be completely random, as long as 0 has some moderate finite chance of occurring.

3. 2^n where n=4.6*10^7 has over 13 million digits. Assuming a probability of 9/10 for the first digit (not being zero)*9/10 for the second, etc. gives an outcome of about 10^(-595,000), which is really unlikely.

 

 

Edited on August 18, 2012, 1:31 pm
  Posted by broll on 2012-08-18 12:38:47

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