A number a is called a cubic residue modulo p if it is congruent to a cube modulo p, that is, if there is a number b such that a ≡ b3 (mod p).
(a) Make a list of all the cubic residues modulo 5, modulo 7, modulo 11, and modulo 13.
(b) Find two numbers a1 and b1 such that neither a1 nor b1 is a cubic residue modulo 19, but a1b1 is a cubic residue modulo 19. Similarly, find two numbers a2 and b2 such that none of the three numbers a2, b2, or a2b2 is a cubic residue modulo 19.
(c) If p ≡ 2 (mod 3), make a conjecture as to which a’s are cubic residues. Prove that your conjecture is correct.
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