[math-fun] Conjecture on Skewes' number
Skewes' number, e^e^e^79, is a notoriously flamboyant upper bound in analytic number theory. As an upper bound, it has since been replaced by much more moderate results. A <a href=" http://garden.irmacs.sfu.ca/?q=op/is_skewes_number_e_e_e_79_an_integer">conjecture</a> on the Open Problem Garden is that Skewes' number is not integral. I am boggled. Of course we know that e^79 is nonintegral because e is not algebraic. (Is there an easier proof?) But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral?
To me it seems odd that anyone would even care (-: On 2012-04-27, Allan Wechsler <acwacw@gmail.com> wrote:
Skewes' number, e^e^e^79, is a notoriously flamboyant upper bound in analytic number theory. As an upper bound, it has since been replaced by much more moderate results.
A <a href=" http://garden.irmacs.sfu.ca/?q=op/is_skewes_number_e_e_e_79_an_integer">conjecture</a> on the Open Problem Garden is that Skewes' number is not integral.
I am boggled. Of course we know that e^79 is nonintegral because e is not algebraic. (Is there an easier proof?)
But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral?
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
That's absolutely right, and it's hard to imagine any technique that could shed light on this supremely useless question. But isn't it a little interesting that even the much more modest questions I asked seem rather hard to answer? On 4/27/12, Robert Munafo <mrob27@gmail.com> wrote:
To me it seems odd that anyone would even care (-:
On 2012-04-27, Allan Wechsler <acwacw@gmail.com> wrote:
Skewes' number, e^e^e^79, is a notoriously flamboyant upper bound in analytic number theory. As an upper bound, it has since been replaced by much more moderate results.
A <a href=" http://garden.irmacs.sfu.ca/?q=op/is_skewes_number_e_e_e_79_an_integer">conjecture</a> on the Open Problem Garden is that Skewes' number is not integral.
I am boggled. Of course we know that e^79 is nonintegral because e is not algebraic. (Is there an easier proof?)
But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral?
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
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Allan Wechsler wrote:
But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral?
Regarding the latter, in Mathematica: $MaxExtraPrecision=Infinity; N[FractionalPart[E^E^10],20] 0.81985800792189826174 e^e^20 is probably doable, but would take some time on my old machine.
Assuming Schanuel's conjecture, one can show that e^e^e^79 is transcendental. Warut On Sat, Apr 28, 2012 at 5:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Skewes' number, e^e^e^79, is a notoriously flamboyant upper bound in analytic number theory. As an upper bound, it has since been replaced by much more moderate results.
A <a href=" http://garden.irmacs.sfu.ca/?q=op/is_skewes_number_e_e_e_79_an_integer">conjecture</a> on the Open Problem Garden is that Skewes' number is not integral.
I am boggled. Of course we know that e^79 is nonintegral because e is not algebraic. (Is there an easier proof?)
But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Allan Wechsler -
Hans Havermann -
Robert Munafo -
Warut Roonguthai