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1 solution
We have 3 3 a + 5 5 b = 1 5 a b 1 2 1 = ( 3 a − 1 1 ) ( 5 b − 1 1 ) then note that 1 2 1 = ( 1 1 ) ( 1 1 ) = ( − 1 1 ) ( − 1 1 ) = ( 1 ) ( 1 2 1 ) = ( − 1 ) ( − 1 2 1 ) . Testing all these possible cases, we arrive that ( a , b ) does not have integer pair solution.
Note: ( a , b ) = ( 0 , 0 ) is not defined.