MathDB

Let

p

be a prime and let l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ . be homogeneous linear polynomials with integral coefficients. Suppose that for every pair

(\xi,\eta)

of integers, not both divisible by

p

, the values

l_k(\xi,\eta), \;1\leq k\leq p^2

, represent every residue class

\textrm{mod} \;p

exactly

p

times. Prove that the set of pairs

\{(a_k,b_k): 1\leq k \leq p^2 \}

is identical

\textrm{mod} \;p

with the set \{(m,n): 0\leq m,n \leq p\minus{}1 \}.

Problem Statement