Problem 6

An ordered pair

(x,y)

of integers is called a primitive point if

\gcd(x,y)=1

. Given a finite set

S

of primitive points, prove that there exist a positive integer

n

and integers

a_0,a_1,\ldots,a_n

such that for every

(x,y)\in S

, we have

a_0x^n+a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}+a_ny^n=1

Step 8 of 8: Verify the new point and finish

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In plain words

Bezout makes the linear factor equal to one at the new point, and the chosen coefficient corrects the remaining value.

Q(x_k,y_k)=P(x_k,y_k)^U+CD=1

Detailed analysis

At $q=(x_k,y_k)$ , the linear form is 1 by $ax_k+by_k=1$ , and the determinant product is $D=\prod_{i=1}^{k-1}d_i$ . Thus $Q(x_k,y_k)=P(x_k,y_k)^U+CD=1$ by the definition of C. Together with the previous step, $Q(x,y)=1\text{ on }S$ . This completes the induction and proves the required statement.