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February 22nd, 2014, 08:22 AM   #11
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Re: FLT otherwise

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by mobel You can not find convergent integer sequences?
There is no apparent meaning of "convergent integer sequence" in mathematics, but from what I can extract from your discussion above, you probably mean :

$\lim_{n \to \infty} A(n)^3 + B(n)^3= C(n)^3$

If so,

$a^3 + b^3= c^3 \pm 1$

Works well. There are infinitely solutions for the above, and all listed in OEIS. It's what we call "near-miss FLT counterexamples" : A050791, A050787
Thank you very much for the reference.
I`m happy now!

February 22nd, 2014, 09:34 AM   #12
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Re: FLT otherwise

Quote:
 Originally Posted by mathbalarka There is no apparent meaning of "convergent integer sequence" in mathematics
I assumed he meant a convergent sequence which was integer-valued, i.e., an eventually constant integer sequence.

Quote:
 Originally Posted by mathbalarka from what I can extract from your discussion above, you probably mean : $\lim_{n \to \infty} A(n)^3 + B(n)^3= C(n)^3$
Perhaps you mean
$\lim_{n\to\infty}A(n)^3+B(n)^3-C(n)^3=0$
? Otherwise you have an unquantified n on the RHS.

February 22nd, 2014, 10:08 AM   #13
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Re: FLT otherwise

Quote:
 Originally Posted by CRGreathouse Perhaps you mean $\lim_{n\to\infty}A(n)^3+B(n)^3-C(n)^3=0$ ? Otherwise you have an unquantified n on the RHS.
Well, I intended lim (A(n)^3 + B(n)^3)/C(n)^3 = 1 originally.

February 22nd, 2014, 10:10 AM   #14
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Re: FLT otherwise

Quote:
 Originally Posted by mathbalarka Well, I intended lim (A(n)^3 + B(n)^3)/C(n)^3 = 1 originally.
No problem, you can get that in lots of different ways. For example A(n) = C(n) = n, B(n) = 1.

February 22nd, 2014, 11:07 AM   #15
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Re: FLT otherwise

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by mobel You can not find convergent integer sequences?
There is no apparent meaning of "convergent integer sequence" in mathematics, but from what I can extract from your discussion above, you probably mean :

$\lim_{n \to \infty} A(n)^3 + B(n)^3= C(n)^3$

If so,

$a^3 + b^3= c^3 \pm 1$

Works well. There are infinitely solutions for the above, and all listed in OEIS. It's what we call "near-miss FLT counterexamples" : A050791, A050787
Ah , yes , thank you very much. I'm surprised i forgot about this , especially the Ramanujan Taxicab Number.

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