Pick one, and really do some exploration and write about it. See Section 7.1 for more information.
Do exploration to try to find a criterion for which primes \(p\) there are square roots of \(-1\). You will have to examine primes less than 10 by hand to make sure you are right!
Do exploration to find out anything you can about how many square roots of \(1\) there are for a given \(n\).
Figure out how many solutions \(x^2\equiv x\) (mod \(n\)) has for \(n=5,6,7\), and then compute how many solutions there are modulo \(210\).
Find solutions to \(x^2+8\equiv 0\) (mod \(121\)) using the method above in Theorem 7.2.3.
Solve \(f(x) = x^3-x^2+2x+1\equiv 0\) (mod \(5^e\)) for \(e=1,2,3\).
Show that Wilson's Theorem fails for \(p=10\) and check that it works for \(p=11\) by computing \(11!\) and then reducing.
In the same setup as in Wilson's Theorem, what is the value of \((j-2)!\), depending upon the modulus?
Use Fermat's Little Theorem to help you calculate each of the following very quickly:
\(512^{372}\) (mod \(13\))
\(3444^{3233}\) (mod \(17\))
\(123^{456}\) (mod \(23\))
Prove Fermat's Little Theorem using the steps in Theorem 7.5.2, or any way you would like.
Prove that Wilson's Theorem always fails if the modulus is not prime. Hint: use the fact that the modulus \(n\) then has factors \(m\) other than \(1\) or \(n\).
Prove that it is impossible for \(p\mid x^2+1\) if a prime \(p\) has \(p\equiv 3\) (mod \(4\)) – that is, if \(p\) is of the form \(4n+3\). (Hard.)
Prove that \(x^2+y^2=p\) has no (integer) solutions for prime \(p\) with that same form.
Show that \(y^2=x^3+999\) has no (integer) solutions.
