It will be shown that some binary quadratic forms
$F_{a,b,c}(x,y)=ax^2+bxy+cy^2$ are def-complete, i.e. they
suffice to define addition and multiplication on the set
$\Bbb N$ of nonnegative integers. Therefore elementary
theories of the corresponding structures are undecidable.
%Sometimes no linear functions are necessary, and the
%quadratic form alone suffices. For example, the operations
$+$, $\times$ are first order definable in the structure
$\langle\Bbb N; F_{1,0,1}\rangle$, where
$F_{1,0,1}(x,y)=x^2+y^2$ for all $x,y\in\Bbb N$. The same
holds for the quadratic forms $F_{6,0,1}$, $F_{15,0,1}$, for
every of the operations $F_{a,-b,c}$ %, $F_{a,d,c}$ provided
$a,b,c\in\NN$ and $0<b^2\le 4ac$,  %and $0<d^2=4ac$, and for
some further quadratic forms.
