In page 2 of Mordell "Diophantion equations" it is given a necessary condition for the solvability of a Diophantine equation:

**Integer solutions of the inhomogeneous equation $f(x) = 0$ can exist only if the congruence
$$f(x) = 0 \pmod M$$
has solutions for all integers $M$.**

Is the converse true?

More stronger, if $$f(x) = 0 \pmod{p^t}$$ for some $p$ and for every $t$, is it true that $f(x)=0$ is soluble? If not could you please give some examples or references for such problems?

More precisely, if the variables in the (multi-variable) Diophantine equation are bounded, it seems we can apply the converse of the Theorem mentioned in the beginning, i.e., for a given and fixed $X$ $$ f(x_1,\ldots,x_s)=0,\qquad |x_i|\leq X $$ it is possible to solve the equation $$ f(x_1,\ldots,x_s)=0\pmod{p^t},\qquad |x_i|\leq X $$ for large enough $p^t$, where $p^t$ is at least some constant multiple of $X^d$, where $d$ is the highest degree of $f(x_1,\ldots,x_s)$.

rationalsolutions. This is true for quadratic equations, but not generally. There are failures for certain cubics (maybe Google-able or in Wiki). $\endgroup$ – paul garrett Aug 31 '16 at 14:07