  One of the most interesting results of the last century was the proof
completed by Matijasevich that computably enumerable sets are precisely the
diophantine sets [MRDP Theorem, 9], thus settling, based on previously
developed machinery, Hilbert's question whether there exists a general
algorithm for checking the solvability in integers of any diophantine equation.
In this paper we describe techniques to prove the nonexistence of polynomials
in two variables for some simple generalizations of the Fibonacci sequence
(explicit diophantine representation of Fibonacci numbers were known from
Jones' polynomial whose positive values have the same range as that of
Fibonacci numbers), and we believe similar techniques exist for the primes. In
this paper we mainly show the following results: (1) using one of the many
techniques known for solving the Pell's equation, namely the solution in an
extended number system, we prove the existence and explicitly find the
polynomials for the recurrences of the form $e(n)=ae(n-1)+e(n-2)$ with starting
values of 0 and 1 in particular, and for any arbitrary starting values, in the
process defining a concept of fundamental starting numbers, (2) we prove a few
identities that seem to be quite interesting and useful, (3) we use these
identities in a novel way to generate systems of equations of certain rank
deficiency using which we disprove for the first time the existence of any
polynomial in 2 variables for the generalized recurrence of the form
$e(n)=ae(n-1)+be(n-2)$
