 mersenneforum.org Rationality of an expression
 Register FAQ Search Today's Posts Mark Forums Read 2018-09-06, 13:02 #1 jnml   Feb 2012 Prague, Czech Republ 110010012 Posts Rationality of an expression Given $$x \in \mathbb{Q}, x \neq 0,$$ can or cannot $$\sqrt{x^3+1} \in \mathbb{Q}$$ be true?   2018-09-06, 13:12 #2 paulunderwood   Sep 2002 Database er0rr 672 Posts Let x=1, then 2^{1/3} is irrational. FLT might have something to do with the stated problem. Last fiddled with by paulunderwood on 2018-09-06 at 13:18   2018-09-06, 13:48 #3 axn   Jun 2003 22·32·151 Posts fermat's last theorem EDIT:- Loophole abuse - x=-1 Last fiddled with by axn on 2018-09-06 at 14:00   2018-09-06, 14:51 #4 LaurV Romulan Interpreter   "name field" Jun 2011 Thailand 101000001010012 Posts Say $$x\in\mathbb{Q}$$ not 0, then $$x$$ can be written $$\frac ab$$, so you have $$\ ^3\sqrt{(\frac ab)^3+1}=\ ^3\sqrt{\frac{a^3}{b^3}+1}=\ ^3\sqrt{\frac{a^3+b^3}{b^3}} = \frac{\ ^3\sqrt{a^3+b^3}}{b}$$ which is rational, so it is $$=\frac mn$$, therefore $$\ ^3\sqrt{a^3+b^3}=\frac{bm}{n}$$, and renaming the last with $$c$$, we have $$\ ^3\sqrt{a^3+b^3}=c$$ or $$a^3+b^3=c^3$$ and as Paul said, we just found a counterexample for FLT. Therefore, that can never be true. Last fiddled with by LaurV on 2018-09-06 at 14:53   2018-09-06, 15:13   #5
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

161110 Posts Quote:
 Originally Posted by axn EDIT:- Loophole abuse - x=-1
and there is no more solution.   2018-09-06, 15:27   #6
jnml

Feb 2012
Prague, Czech Republ

3·67 Posts Quote:
 Originally Posted by LaurV ... which is rational ...
Did you mean irrational?   2018-09-06, 15:47   #7
paulunderwood

Sep 2002
Database er0rr

106118 Posts Quote:
 Originally Posted by jnml Did you mean irrational?
It is presumed rational until the contradiction by FLT is established.   2018-09-06, 15:57   #8
jnml

Feb 2012
Prague, Czech Republ

3×67 Posts Quote:
 Originally Posted by paulunderwood It is presumed rational until the contradiction by FLT is established.
I see, thanks. I did not assume that particular implicit assumption ;-)  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post MattcAnderson Homework Help 5 2016-11-01 08:16 1260 Software 17 2015-08-28 01:35 ixfd64 Programming 2 2009-03-01 06:19 tinhnho Miscellaneous Math 4 2005-01-17 19:45 jinydu Puzzles 9 2004-04-02 01:03

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