Question

For which primes p is 15 a square modulo p?

Answer #1

Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Number Theory:
Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Let p and q be primes. Prove that pq + 1 is a square if and only
if p and q are twin primes. (Recall p and q are twin primes if p
and q are primes and q = p + 2.) (abstract algebra)

Find all primes p such that p | (a^37−a) for all a ∈N. Multiply
those primes together to ﬁnd the largest n ∈N such that n | (a^37
−a) for all a ∈N.

Let p be a prime and let a be a primitive root
modulo p. Show that if gcd (k, p-1) = 1, then b≡ak (mod
p) is also a primitive root modulo p.

Let p be an odd prime.
Prove that −1 is a quadratic residue modulo p if p ≡ 1 (mod 4),
and −1 is a quadratic nonresidue modulo p if p ≡ 3 (mod 4).

Find the value of the Legendre symbol (−2/p), depending on the
congruence class of p modulo 8.

Find the primes for which 11 is a quadratic
residue.

Suppose G is a group of order pq such p and q are primes, p<q
and therefore |H|=p and |K|= q, where H and K are proper subgroups
are G. It was determined that H and K are abelian and G=HK. Show
that H and K are normal subgroups of G without using
Sylow's Theorem.

For each prime number p below, find all of the Gaussian primes q
such that p lies below q:
2
3
5
Then for each Gaussian prime q below, find the prime number p
such that q lies above p:
1 + 4i
3i
2 + 3i

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