MathDB
P 18

Source:

May 25, 2007
modular arithmeticnumber theoryAdditive Number Theory

Problem Statement

Let pp be a prime with p≡1(mod4)p \equiv 1 \pmod{4}. Let aa be the unique integer such that p=a2+b2,  a≡−1(mod4),  b≡0  (mod2)p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2} Prove that ∑i=0p−1(i3+6i2+ip)=2(2p),\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right), where (kp)\left(\frac{k}{p}\right) denotes the Legendre Symbol.
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