Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ 0 . However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ 0.

We consider the theory and its weak fragments in the language of arithmetic expanded with the functional symbol . We prove that and its weak fragments, down to and , are subject to the Tennenbaum phenomenon with respect to , , and . For the last two theories it is still unknown if they may have nonstandard recursive models in the usual language of arithmetic.