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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