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 7 of 8: Verify the old points

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In plain words

At every old point the determinant product is zero.

Q(x_i,y_i)=1

Detailed analysis

For an old point $(x_i,y_i)$ , its own factor $y_ix-x_iy$ vanishes. Since $P(x,y)=ax+by$ has value 1 there, the first term of $Q(x,y)=P(x,y)^U+C(ax+by)^V\prod_{i=1}^{k-1}(y_ix-x_iy)$ is 1 and the correction term is 0. Hence $Q(x_i,y_i)=1$ .