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# special type of representation

Thu 24 May, 2012 02:07 am
Find a positive integer m such that 8^m can be represented as ;
8^m = 31 (( 2 n - 1 )^2) + (( 2 w - 1 )^2 ) , where n and w are positive integers
not necessarily distinct.
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raprap

1
Thu 24 May, 2012 11:26 am
@uvosky,
Taking
8^m = 31 (( 2 n - 1 )^2) + (( 2 w - 1 )^2 )
making it
2^(3m) = 31 (( 2 n - 1 )^2) + (( 2 w - 1 )^2 )
using
k=( 2 n - 1 )^2) + ( 2 w - 1 )^2
this becomes
2^(3m)=31k
k=2^(3m)/31 where k is an integer
as 31 is prime and not divisible by 2
I don't think there is an integral solution.

Rap
markr

1
Thu 24 May, 2012 02:22 pm
@raprap,
There is no 31*(2w-1)^2 term in the original equation.
raprap

1
Thu 24 May, 2012 02:44 pm
@markr,
Oops--miscounted the brackets

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