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 5 of 8: Choose compatible degrees

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In plain words

Euler’s theorem supplies a power congruent to one modulo the determinant product.

V=nU-(k-1)

Detailed analysis

Because $P(x_k,y_k)$ is coprime to $D=\prod_{i=1}^{k-1}d_i$ , choose a sufficiently large positive integer $U$ divisible by the exponent supplied by Euler's theorem so that $P(x_k,y_k)^U\equiv1\pmod{|D|}$ . Enlarge it if necessary to ensure $nU\ge k-1$ . Define $V=nU-(k-1)$ ; then V is a nonnegative integer.