a = 12|[32876712]^k b = [32876712]^k|33 => a^2 + b^2 = a|b Examples 12^2 + 33^2 = 1233 1232876712^2 + 3287671233^2 = 12328767123287671233 &c
Cute! How do you prove this? How did you notice it? Jim Propp On Friday, September 7, 2018, David Wilson <davidwwilson@comcast.net> wrote:
a = 12|[32876712]^k b = [32876712]^k|33 => a^2 + b^2 = a|b
Examples
12^2 + 33^2 = 1233 1232876712^2 + 3287671233^2 = 12328767123287671233 &c
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JP: "Cute! How do you prove this?" See Steven Charlton's Self-Similar Sums of Squares: https://www.math.uni-tuebingen.de/user/charlton/writings/squaresumcat/sums_o...
Also, here's a length to Alf van der Poorten's paper on generalizations of this (the link in Charlton's paper doesn't work). Victor https://www.researchgate.net/profile/Michael_Kleber/publication/225591088_A_... On Sat, Sep 8, 2018 at 10:32 AM Hans Havermann <gladhobo@bell.net> wrote:
JP: "Cute! How do you prove this?"
See Steven Charlton's Self-Similar Sums of Squares:
https://www.math.uni-tuebingen.de/user/charlton/writings/squaresumcat/sums_o...
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I once saw the equation 588^2 + 2353^2 = 5882353 and immediately recognized the relationship to the decimal fraction 1/17 = .[0588235294117647] Following up on that observation resulted in discovery of a general method for generating a family of solutions to a^2 + b^2 = a|b based on certain decimal fractions. This method produced the family 12^2 + 33^2 = 1233 1232876712^2 + 3287671233^2 = 12328767123287671233 &c related to the fraction 9/73 = .[12328767]* and of course there are an infinitude of other examples. Given your level of mathematical accomplishment, I'm sure you'll enjoy reconstructing my reasoning and method. It might be worth a paper in a recreational math journal, but I'm not competent to write one.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Friday, September 07, 2018 10:47 PM To: math-fun Subject: Re: [math-fun] Oddity
Cute! How do you prove this? How did you notice it?
Jim Propp
On Friday, September 7, 2018, David Wilson <davidwwilson@comcast.net> wrote:
a = 12|[32876712]^k b = [32876712]^k|33 => a^2 + b^2 = a|b
Examples
12^2 + 33^2 = 1233 1232876712^2 + 3287671233^2 = 12328767123287671233 &c
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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David Wilson -
Hans Havermann -
James Propp -
Victor Miller