mersenneforum.org - Less than 4/3

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XYYXF

2008-12-02 21:50

Less than 4/3

How to prove without a calculator that

[TEX]\sqrt[12]{2}\sqrt[7]{5}[/TEX]

is less than 4/3?

Uncwilly

2008-12-02 23:02

[QUOTE=XYYXF;151708]How to prove without a calculator that[/QUOTE]Paper and pencil and successive aproximation.

Batalov

2008-12-03 02:48

take 84-th power of both sides:
374141099455063118393835625834907722052531250000000 <
374144419156711147060143317175368453031918731001856 :-)

cheesehead

2008-12-03 03:31

[quote=Batalov;151730]374144419156711147060143317175368453031918731001856 :-)[/quote]The 84th power of 4/3 is an even integer?

Oh, wait -- there's a 3^84 factor in the other one, isn't there? Let's see if Factoris ([url]http://wims.unice.fr/wims/wims.cgi[/url]) handles that.

Yep: "374141099455063118393835625834907722052531250000000 = 2[sup]7[/sup] [COLOR=green]×[/COLOR] 3[sup]84[/sup] [COLOR=green]×[/COLOR] 5[sup]12[/sup]"

davieddy

2008-12-03 13:03

I find it difficult to believe a calculator has not been used
in this proof.

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