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\centerline{\bf THEORY OF NUMBERS, Math 115 B}
\centerline{\bf Homework 2}
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\begin{enumerate}

\item Prove that for the area $S(M)$ of a simple lattice polygon $M$
the following inequality holds:

   $$S(M) \geq G -L/2-1$$

where $G$ denotes the total number of lattice nodes lying inside $M$
(i.e. $G=i+b$), $L$ denotes the perimeter of the polygon (i.e. the length
of its boundary). Can you think of a case when equality holds?

\item Construct a set $C$ in the plane that is convex, symmetric about $0$
and has area 4, but contains no nonzero integral point.

\item Construct a set $C$ that is convex and has infinite area but contains
no integral point.

\item Can you write every non-negative integer as a sum of three squares?

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