Quadratic Residues and Non-residues Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Find all of the quadratic residues of the integer 7.

(2) Find all of the quadratic residues of the integer 8.

(3) Find all of the quadratic residues of the integer 15.

(4) Find all of the quadratic residues of the integer 18.

(5) Find the value of the Legendre symbol for and 6.

(6) Evaluate the Legendre symbol using Euler's criterion.

(7) Evaluate the Legendre symbol using Gauss's lemma.

(8) Let and be integers not divisible by the prime Show that either one or all three of the integers and are quadratic residues of

(9) Find all solutions of the quadratic congruence

(10) Find all solutions of the quadratic congruence

(11) Find all solutions of the quadratic congruence

(12) Find all solutions of the quadratic congruence

(13) Find all solutions of the quadratic congruence

(14) Find all solutions of the quadratic congruence   

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