Math 345/645 - Weekly homework 8
The problems on this assignment should be written up impeccably and turned in on Tuesday,
April 12. This assignment needs to be typed in LATEX. A problem with a ∗ is extra-credit for
undergraduates and required for graduate students. A problem with a ∗∗ is extra-credit for
undergraduates and graduate students.
1. Suppose that n = x2 is a perfect square with an even number of base-10 digits. Assume
that when n is written backwards, you get another perfect square y 2 . Prove that 121|n. [
Forexample, 1089 = 332 and 9801 = 992 . Use the mod 11 divisibility test, and the fact that
−1
= −1.]
11
2. Let p > 2 be an odd prime, and assume there is some x ∈ Z so that ordp (x) = p − 1. (That
such an x exists follows from Theorem 6.8, which we didn’t prove.) Use this assumption to
reprove Theorem 7.2 and 7.3.
3. ∗ Prove that there are infinitely many primes p ≡ 11 (mod 12). [Hints: Assume there are
finitely many and let N be their product. Show that there is a prime divisor p|3N 2 − 4 with
p ≡ 11 (mod
( 12) and derive a contradiction. You will want to use the fact, proven in class,
1
if p ≡ 1 or 11 (mod 12)
that p3 =
]
−1 if p ≡ 5 or 7 (mod 12).
4. ∗∗ Attend the colloquium on Wednesday, April 6 about the distribution of prime numbers.
Say that you attended the colloquium, and write something (fairly short) about something you
learned at it.
1