Assignment # 6 – Number Theory

(1) Evaluate the following Legendre symbols:
(a) (11/29)
(b) (23/61)

(2) Let p > 5 be a prime. Show that at least one of 2, 5, 10 is quadratic residue modulo
p.
(3) Show that for p > 5 a prime, there are always consecutive integers that are quadratic
residues modulo p, and consecutive integers that are quadratic nonresidues modulo
p. (Hint: use the previous question.)
(4) Compute the product of all the quadratic residues a where (a; p) = 1 in a residue
system modulo p where p is prime. Similarly, compute the product of all the quadratic
nonresidues in a residue system modulo p.
(5) Show that the smallest positive quadratic nonresidue modulo p (where p is a prime)
is always a prime.
(6) Use quadratic reciprocity to evaluate the following Legendre symbols. Justify your
answers.
(a) (22/37)
(b) (31/61)

assignment 6 solution