Evaluate 311 three ways 1 Computing the squares modulo 11 2

Evaluate (3/11) three ways: (1) Computing the squares modulo 11, (2) Euler\'s Criterion, and (3) Gauss\' Lemma.

Solution

(1)

1^1 = 1 mod 11

2^2 = 4 mod 11

3^2= 9 mod 11

4^2= 16=6 mod 11

5^2= 25=3 mod 11

Note that: a^2=(p-a)^2 mod 11. So we don\'t need to compute any further

5^2=3 mod 11.

Hence, (11-5)^2=6^2=3 mod 11

So two solutions to: x^2=3 mod 11

Hence,

(3/11)=1

(2)

By Euler\'s criterion

(3/11)=3^{(11-1)/2} mod 11 = 3^5 mod 11=243 mod 11=22*11+1=1 mod 11

Hence, (3/11)=1

(3)

We compute:

3,2*3,.....,(11-1)/2*3

and reduce them to least residue modulo 11 and count how many are larger than 11/2

3,6,9,12,15

Reducing them to least residue modulo 11

3,6,9,1,4

Numbers larger than:11/2=5.5 are

6,9 = 2 numbers

Hence,

(3/11)=(-1)^2=1