[math-fun] base 12 things and nothing happened today at 12:12:12
Hello, believe it or not, there are somepeople that are convinced that a system in base 12 would be better, for reading the hour and some trigonometric reasons, there is even a film aboutit, some aliens with 12 toes, that came out in the newspaper 'le monde' since today at 12:12:12 andon year 12, nothing happened, here is the film : http://youtu.be/B6415P4bh5M and here is a link to the Duodecimal Association, http://www.dozenal.org/index.html they call themselves Dozenalist, I presume they must like eggs a lot? best regards, Simon Plouffe
And OMG it was just noon here!!!!!!!!!!!! Anyway, as Ramanujan is famous for pointing out to Hardy, 9^3 + 10^3 = 12^3 + 1 . This is a counterexample to Fermat's Last Theorem . . . almost. It's off by one. QUESTION: Are there other examples of integers x, y, z for which x^3 + y^3 = z^3 + 1 ??? (Does the restatement (x-1)(x^2 + x + 1) = (z-y)(z^2 + zy + y^2) help?) *Positive* integers x, y, z ??? And how about for integer exponents > 3 ??? --Dan
Just wrote a little program and ran it on numbers <= 1000, to get: << 9^3 + 10^3 = 12^3 + 1 64^3 + 94^3 = 103^3 + 1 73^3 + 144^3 = 150^3 + 1 135^3 + 235^3 = 249^3 + 1 244^3 + 729^3 = 738^3 + 1 334^3 + 438^3 = 495^3 + 1
Hmm, now I wonder about the asymptotic density of solutions. --Dan I wrote:
QUESTION: Are there other examples of integers x, y, z for which
x^3 + y^3 = z^3 + 1
???
See https://sites.google.com/site/tpiezas/010 for the parametric solutions. Warut On Thu, Dec 13, 2012 at 4:34 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Just wrote a little program and ran it on numbers <= 1000, to get:
<< 9^3 + 10^3 = 12^3 + 1
64^3 + 94^3 = 103^3 + 1
73^3 + 144^3 = 150^3 + 1
135^3 + 235^3 = 249^3 + 1
244^3 + 729^3 = 738^3 + 1
334^3 + 438^3 = 495^3 + 1
Hmm, now I wonder about the asymptotic density of solutions.
--Dan
I wrote:
QUESTION: Are there other examples of integers x, y, z for which
x^3 + y^3 = z^3 + 1
???
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This is sequence A050791 in OEIS, where you will find many more terms and references Neil On Wed, Dec 12, 2012 at 4:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Just wrote a little program and ran it on numbers <= 1000, to get:
<< 9^3 + 10^3 = 12^3 + 1
64^3 + 94^3 = 103^3 + 1
73^3 + 144^3 = 150^3 + 1
135^3 + 235^3 = 249^3 + 1
244^3 + 729^3 = 738^3 + 1
334^3 + 438^3 = 495^3 + 1
Hmm, now I wonder about the asymptotic density of solutions.
--Dan
I wrote:
QUESTION: Are there other examples of integers x, y, z for which
x^3 + y^3 = z^3 + 1
???
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Dec 12, 2012 at 10:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Just wrote a little program and ran it on numbers <= 1000, to get:
<< 9^3 + 10^3 = 12^3 + 1
64^3 + 94^3 = 103^3 + 1
73^3 + 144^3 = 150^3 + 1
135^3 + 235^3 = 249^3 + 1
244^3 + 729^3 = 738^3 + 1
334^3 + 438^3 = 495^3 + 1
Seeing that some of the solutions contain already perfect squares or prime powers, you could also ask about the solutions of the kind p^6 + y^3 = z^3+1 (example Ramanujan) x^18 + y^3 = z^3+ 1 (example 64,94,103 or 729,244,738) and other variants. Olivier
Sure, 334^3 + 438^3 = 495^3 + 1. On 12/12/12, Dan Asimov <dasimov@earthlink.net> wrote:
And OMG it was just noon here!!!!!!!!!!!!
Anyway, as Ramanujan is famous for pointing out to Hardy,
9^3 + 10^3 = 12^3 + 1
.
This is a counterexample to Fermat's Last Theorem . . . almost. It's off by one.
QUESTION: Are there other examples of integers x, y, z for which
x^3 + y^3 = z^3 + 1
???
(Does the restatement (x-1)(x^2 + x + 1) = (z-y)(z^2 + zy + y^2) help?)
*Positive* integers x, y, z ???
And how about for integer exponents > 3 ???
--Dan
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-- Charles Greathouse Analyst/Programmer Case Western Reserve University
Noam Elkies, in http://arxiv.org/pdf/math/0005139v1.pdf gave an algorithm to find all solutions to |x^3 + y^3 - z^3| < M with |x|,|y|,|z| <= N in time (log^O(1) N) M, among others. One his solutions appeared on a Simpson's episode! Victor On Wed, Dec 12, 2012 at 4:52 PM, Hans Havermann <gladhobo@teksavvy.com>wrote:
Dan Asimov:
x^3 + y^3 = z^3 + 1
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participants (8)
-
Charles Greathouse -
Dan Asimov -
Hans Havermann -
Neil Sloane -
Olivier Gerard -
Simon Plouffe -
Victor Miller -
Warut Roonguthai