PMATH 340 Final
PMATH代写 This is the fifinal exam for PMATH 340. Ideally, this test will not take more than 3-4 hours but that is not a hard time limit.
1 Preface
This is the fifinal exam for PMATH 340. Ideally, this test will not take more than 3-4 hours but that is not a hard time limit. You are free to open this test at 5PM on the 12th and not start doing any work until Saturday morning if you so desire. You can work on this when you want and take breaks as you need. The material in this exam is the material from the whole class, and is similar to material discussed in the homework, the practice problems, and the problem sessions. As stated in the syllabus, I aim for this to be easier than the homework. There should be fewer ugly answers and things in general should “work out” nicely.
The submission on crowdmark will close on Saturday, August 14th at 4:59PM EST. I implore you to have Crowdmark open at least 30 minutes before then and anticipate the uploading process to take some time. I also strongly suggest you double check to make sure your fifiles got uploaded. If, for any reason, anything goes wrong, please email me by 4:45PM EST telling me so and attaching pictures of your solutions. I am far more sympathetic to people who have evidence that they fifinished things before the deadline.
This test will be open book and open notes. Feel free to look things up in the book and the recorded problem sessions. However, I require that you avoid using other materials (wikipedia, other textbooks, and so on). You are free to use a four-function calculator, although I have designed this test so that shouldn’t be necessary. PMATH代写
This exam is meant to be your work and your work alone. Please do not discuss material on this exam with other people (especially other students in this class) until Saturday at 5PM EST. If you have any questions about what things mean or need anything clarifified, please email me or contact me on Piazza. If you email a TA, they will immediately forward the email to me. Additionally, if I need to clarify anything, I will make a post on Learn that has all clarififications that I have told people. I will not answer correctness questions nor will I give hints for this exam.
I will try to be as available as possible during this 48-hour window, but I do have a couple of prior commitments (most notably I will at least attempt to sleep for 8 hours each night). I will respond to emails and Piazza posts as promptly as I am able to.
Finally, I implore you to not cheat. In time, the details of this specifific grade will be washed away, but whether you learned this material and understood it will remain. On a serious note, if I suspect someone of cheating, I may audit them: ask to have a meeting to discuss a specifific answer to a question on short notice. If they do poorly their grade may be reduced. If they don’t show up, their grade will defifinitely be reduced. I sincerely hope that I don’t have to do this, but I understand that there does need to be the threat of punishment for cheating.
Best of luck, and happy solving!
2 The Problems PMATH代写
Problem 1. For each of the following equations (mod 35), determine if there are solutions to the equations. If so, fifind all of them and if not explain why not.
(a) 3x ≡ 31 (mod 35)
(b) 20x ≡ 25 (mod 35)
(c) 21x ≡ 15 (mod 35)
(d) x2 ≡ 11 (mod 35)
(e) x2 ≡ 13 (mod 35)
Problem 2. This problem asks various questions about √ 29.
(a) For each odd prime p in the interval [30, 50], determine whether 29 is a square (mod p).
(b) Find the continued fraction expansion of √ 29.
(c) Find a solution in integers x and y to x2 − 29y2 = 1 with y≠0 PMATH代写
Problem 3.
(a) Without calculating the value of M83 = 283 − 1, show that 167|M83.
(b) Assume that p and 2p + 1 are both prime (if this happens, p is said to be a Sophie Germain prime). Give (with proof) a criterion on p such that (2p + 1)|(2p − 1).
Problem 4. The pentagonal numbers, written Pn, are defifined by counting the number of points in the following diagrams:
(a) Find a closed form for the pentagonal numbers.
(b) Let x and y be integers. Find variables X and Y in x and y respectively, as well as an integer D such that a solution to Px = y2 gives rise to a solution to X2 − DY 2 = 1.1
1See the discussion at the start of chapter 31 of the book if what is meant by this if this problem is unclear.
2(c) Find a way to classify all such solutions to X2 −DY 2 = 1, and use that to fifind all pentagonal numbers that are also squares.
Problem 5. The cyclotomic polynomials, written Φn(x) are defifined by the recurrence relation Φd(x) = xn − 1. As a couple examples, one has that Φ1(x) = x − 1, Φ2(x) = x + 1, and Φ3(x) = x2 + x + 1. Throughout this problem, you may assume that these are polynomials with integer coeffiffifficients and not just rational functions over Q.
Compute (with justifification) the following:
(a) Φn(x) for 4 ≤ n ≤ 10,
(b) Φn(0) and Φn(1) for all n, and
(c) deg(Φn(x)) for all n.
Problem 6. Recall that a positive integer is called abundant if σ(n) > 2n. PMATH代写
(a) Show that, if a|b, then σ(aa) ≤ σ(bb) , with equality if and only if a = b.
(b) Show that there are 12 elements of Z/60Z such that, if n > 6 is an integer congruent to one of those elements, then n is abundant.
This shows that at least 20% of all numbers are abundant.
Problem 7. This problem will handle a couple more cases of Dirchelet’s Theorem.
(a) Show that, if p is an odd prime such that p|(x4 + 1) for some integer x, then p ≡ 1 (mod 8).
(b) Show that there are infifinitely many primes congruent to 1 (mod 8).
(c) Upgrade your argument above to show that there are infifinitely many primes that are congruent to 9 (mod 16).
