MATH 319 Quadratic Residues

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1. It was proved in class that for an odd prime p, different from 7, if and only if Find a prime p of each of the forms and find a solution to .

a)

b)

c)

d)

e)

f)

2. In the numtheory library in Maple, there is a function Legendre(a,p) which computes the Legendre symbol, for odd primes p. For example, to compute the Maple syntax is Legendre(7,11). Using Maple fill in the following table for the first 18 primes. Note that you are asked to compute both for each of the primes.

p			p			p
3			19			43
5			23			47
7			29			53
11			31			59
13			37			61
17			41			67

Guess a theorem about the primes for which 2 is a quadratic residue.

3. In this question you are asked to come up with congruence conditions on odd primes p (different from 11) such that . The condition we found in class for used p(mod 12) while that for used p(mod 28). What would be a good modulus m for 11? m = _______

p			p			p
3			29			61
5			31			67
7			37			71
11			41			73
13			43			79
17			47			83
19			53			89
23			59			97

_______________________(mod m)