<i>"But the assumption that digits in a big power of two occur at random"</i><p>That may not be the best of assumptions, but it doesn't hurt his argument. Benson's law learns us that the initial digits of 2^n tend to be low (empirically, but in this case, also mathematically. See <a href="http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2004-09.html" rel="nofollow">http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2004-...</a>). Less than 8% of powers of two begins with the digit '5'. Also, fewer powers of five begin with an even digit than with an odd digit.<p>The argument that there <i>is</i> no deep mathematical reason why it has to be true, though, I find more belief than math. We don't know such a reason, but that doesn't mean it doesn't exist.