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I came across this problem and i could not solve it despite many tries. Show that $3^{2008}$ + $4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$. I think the question can be solved more easily..Can anybody suggest any other method.

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    $\begingroup$ Hi -- welcome to math.SE! Here's a reference and tutorial for typesetting math on this site. $\endgroup$
    – joriki
    May 30, 2016 at 8:56
  • $\begingroup$ well no i am not doing so.............tatan $\endgroup$
    – Pole_Star
    Jun 1, 2016 at 10:38
  • $\begingroup$ may be.... may be not...besides i m new to these things ..i might do something wrong $\endgroup$
    – Pole_Star
    Jun 1, 2016 at 12:59

4 Answers 4

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Using $a^2+b^2 =(a+b)^2-2ab$, we have

$3^{2008} +4^{2009}=(3^{1004})^2+(2^{2009})^2=(3^{1004}+2^{2009})^2 - 2 \cdot 2^{2009} \cdot 3^{1004}$

So we have $3^{2008} +4^{2009}=(3^{1004}+2^{2009})^2- 2^{2010} \cdot 3^{1004}$

Now, using $a^2-b^2=(a-b)(a+b)$, we have

$3^{2008} +4^{2009}=(3^{1004}+2^{2009}+2^{1005} \cdot 3^{502})(3^{1004}+2^{2009}-2^{1005} \cdot 3^{502})$


Finally, it is not hard to see that $(3^{1004}+2^{2009}-2^{1005} \cdot 3^{502}) > 2009^{182}$, which is left as an exercise to the reader.

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  • $\begingroup$ Superb answer ! $\endgroup$
    – Peter
    May 30, 2016 at 9:36
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    $\begingroup$ Yes, great answer. The reader exercise:$$2^{2009} = 2^{2008} +2^{2008} \\2^{2008} = 2^{1005}2^{1003} > 2^{1005}2^{808} >2^{1005}3^{505} >2^{1005}3^{502}\\ \text{and } 2^{2008} > 2^{2002} = (2^{11})^{182}>2009^{182}$$ $\endgroup$
    – Joffan
    Jan 14, 2017 at 15:45
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Hint:-

$$ a^4+4b^4=(a^2+2b^2-2ab)(a^2+2b^2+2ab)$$(Sophie Germain's Identity)

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Hint:-

We try to factorise the equation and show that each factor is greater than $2009^{182}$.

$$3^{2008}+4^{2009}=(\color{green}{3^{502}})^\color{orange}{4}+\color{orange}{4}\times\color{blue}{(4^{502})}^\color{orange}{4}.$$

Now,interestingly,this is of the form of Sophie-Germain's identity which states-

$$a^4+4b^4=(a^2+2ab+2b^2)(a^2-2ab+2b^2).$$

Take,$(3^{502})=a $ and $4^{502}=b$ and resolve it into two factors and show that each factor is greater than the given value of $2009^{182}$.

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We use the standard factorisation:

x4 + 4y4 = (x2 + 2xy + 2y2)( x2 - 2xy + 2y2).

We observe that for any integers x, y,

x2 + 2xy + 2y2 = (x + y)2 + y2 ≥ y2,

x2 - 2xy + 2y2 = (x - y)2 + y2 ≥ y2,

using sophie germains identity we factorise and write

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