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Find four prime numbers less than 100 which are factor of 3\[
3^{32}-2^{32}\]

\[3^{32}-2^{32}\]
\[=(3^{16}+2^{16})(3^{16}-2^{16})\]
$now,$
\[=3^{16}-2^{16}\]
\[=(3^{8}+2^{8})(3^{8}-2^{8})\]
$now,$
\[3^{8}-2^{8}\]
\[=(3^4+2^4)(3^4-2^4)\]
\[=97 \cdot 65\]
\[=97 \cdot 5 \cdot 13\]

the next thing i did isn't quite successful

\[3^8+2^8\]
\[=17 \cdot 401\]

so, the answers are
\[5,97,13,17\]

Thnx for the answer......

@ataher.sams

i think you didn't notice how big mistakes i did on that post. i didn't even find out $4$ primes.now it's edited and quite fulfilled.

I didn't see it... 'cause I was busy then....

could anyone post a better way to functionize
\[2^8+3^8\]
i just used trial and error division process

Why don't you start a new topic on this??

ataher.sams wrote: ↑

Wed Nov 30, 2011 6:39 pm

Find four prime numbers less than $100$ which are factors of \[
3^{32}-2^{32}\]

This is BdMO National Junior $2008$ problem no $8$.