MathDB

p=3k+1

is a prime number. For each

m \in \mathbb Z_p

, define function

L

as follow:

L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )

a) For every

m \in \mathbb Z_p

and

t \in {\mathbb Z_p}^{*}

prove that

L(m) = L(mt^3)

. (5 points)
b) Prove that there is a partition of

{\mathbb Z_p}^{*} = A \cup B \cup C

such that

|A| = |B| = |C| = \frac{p-1}{3}

and

L

on each set is constant. Equivalently there are

a,b,c

for which

L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.

. (7 points)
c) Prove that

a+b+c = -3

. (4 points)
d) Prove that

a^2 + b^2 + c^2 = 6p+3

. (12 points)
e) Let

X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}

, show that

X,Y \in \mathbb Z

and also show that :

p= X^2 + XY +Y^2

. (2 points)
(

{\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}

)

Problem Statement