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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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The Geometry of Numbers

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#1 h May 25, 2007, 12:25 AM • 2 Y

Y by Adventure10, Mango247

A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$ .

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#2 h Aug 8, 2007, 3:39 AM • 2 Y

Y by Adventure10, Mango247

Pick's Theorem trivializes the problem ( doesn't it?)

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#3 h Mar 7, 2008, 9:49 PM • 2 Y

Y by Adventure10, Mango247

Method 1.
is by simple application of Pick theorem.
$A(\Delta)= i(\Delta) + \frac{1}{2} b(\Delta) - 1$
so
$A(\Delta)= \frac{3}{2} -1= \frac{1}{2}$

Method 2
is as follows:
There is a triangle $\Delta OPQ$ with $O(0,0)$ , $P(x_1,y_1)$ and $Q(x_2,y_2)$ .
We know that points $P(x_1,y_1)$ and $Q (x_2,y_2)$ are visible from the origin $O(0,0)$ .
So, $(x_1, y_1)=1$ , $(x_2,y_2)=1$ . That means that $\frac{x_1}{y_1}$ , and $\frac{x_2}{y_2}$ are irreductible.
So, $\frac{x_1}{y_1}$ and $\frac{x_2}{y_2}$ are the terms of Farey sequence.
But if $\frac{x_1}{y_1}$ , and $\frac{x_2}{y_2}$ are two successive terms of Farey sequence, then
$x_1 \cdot y_2 - x_2 \cdot y_1=1$ .
From the other part a surface of an arbitrary trangle is given by the formula
$A(\Delta)= \frac{1}{2} \mid x_1 \cdot y_2 - x_2 \cdot y_1 \mid$
so
$A(\Delta OPQ ) = \frac{1}{2}$ and it is done.

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