Comments on: Patterns in the Last Digits of the Positive Powers of Two http://www.exploringbinary.com/patterns-in-the-last-digits-of-the-positive-powers-of-two/ Binary Numbers, Binary Code, and Binary Logic Mon, 30 Jan 2012 01:37:52 +0000 hourly 1 http://wordpress.org/?v=3.2.1 By: Rick Regan http://www.exploringbinary.com/patterns-in-the-last-digits-of-the-positive-powers-of-two/comment-page-1/#comment-4458 Rick Regan Wed, 15 Dec 2010 19:53:59 +0000 http://www.exploringbinary.com/?p=249#comment-4458 David, Yes, <a href="http://www.exploringbinary.com/cycle-length-of-powers-of-two-mod-powers-of-ten/" title="Read Rick Regan's Article “Cycle Length of Powers of Two Mod Powers of Ten”" rel="nofollow">my proof</a> is a little long, though I was happy the way it turned out (it took quite a bit of work to do). If your proof is simpler, I'd love to see it, or at least some more of the details. BTW, would your technique work for proving the <a href="http://www.exploringbinary.com/cycle-length-of-powers-of-five-mod-powers-of-ten/" title="Read Rick Regan's Article “Cycle Length of Powers of Five Mod Powers of Ten”" rel="nofollow">cycle length of powers of <em>five</em> mod powers of ten</a>? I rather liked how I did that one, with the recursive factoring of differences of squares. David,

Yes, my proof is a little long, though I was happy the way it turned out (it took quite a bit of work to do). If your proof is simpler, I’d love to see it, or at least some more of the details.

BTW, would your technique work for proving the cycle length of powers of five mod powers of ten? I rather liked how I did that one, with the recursive factoring of differences of squares.

]]> By: David Bandel http://www.exploringbinary.com/patterns-in-the-last-digits-of-the-positive-powers-of-two/comment-page-1/#comment-4457 David Bandel Wed, 15 Dec 2010 18:38:52 +0000 http://www.exploringbinary.com/?p=249#comment-4457 I also noticed the digits of the powers of two cycled with a period growing by a factor of five per digit. Never thought to google the result. I proved it as well but I did it more simply using group theory. Starting from the assumption that the period didn't multiply by five per digit I arrived at a non-commutative cyclic group. But we know that all cyclic groups are Abelian. I also noticed the digits of the powers of two cycled with a period growing by a factor of five per digit. Never thought to google the result.

I proved it as well but I did it more simply using group theory. Starting from the assumption that the period didn’t multiply by five per digit I arrived at a non-commutative cyclic group. But we know that all cyclic groups are Abelian.

]]> By: Carnival of Mathematics #59 « The Number Warrior http://www.exploringbinary.com/patterns-in-the-last-digits-of-the-positive-powers-of-two/comment-page-1/#comment-4240 Carnival of Mathematics #59 « The Number Warrior Fri, 06 Nov 2009 07:08:18 +0000 http://www.exploringbinary.com/?p=249#comment-4240 [...] Rick Regan of Exploring Binary finds patterns in the last digits of the positive powers of two. [...] [...] Rick Regan of Exploring Binary finds patterns in the last digits of the positive powers of two. [...]

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