MathDB

A2 B4 C1

Part of STEMS 2021-22 Math Cat A-B

Problems(1)

Another strange representation of a prime

Source: STEMS 2022 Math Cat A P3/Cat B P4/Cat C P1

1/24/2022
If there are integers a,b,ca,b,c such that a2+b2+c2−ab−bc−caa^2+b^2+c^2-ab-bc-ca is divisible by a prime pp such that gcd(p,a2+b2+c2−ab−bc−cap)=1\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1, then prove that there are integers x,y,zx,y,z such that p=x2+y2+z2−xy−yz−zxp=x^2+y^2+z^2-xy-yz-zx.
number theory