Math 105

Topics in Number Theory

Instructor: Carl Pomerance (carl.pomerance at dartmouth.edu)

News and current assignment

Wheelock Books has informed me that the text is now in stock.

We meet MWF at 11:15, and in 343 Kemeny. (PLEASE NOTE THE NEW ROOM)

Homework due Wednesday, November 9:
Ch. 6, #3.

Abstract

Primes and Polynomials:
This course will develop elementary estimates on the distribution of prime numbers, including elementary sieve methods. A recurrent theme will be the prime factorization of integer-polynomial values. We will follow the first 3 chapters of Pollack's "Not always buried deep", as well as Chapter 6, and also Chapter 3 of Montgomery & Vaughan's "Multiplicative number theory". I recommend acquiring Pollack's book, the second book is optional. Prerequisites: An undergrad number theory course, as well as some abstract algebra. I'll be happy to try and fill in gaps for motivated students. Grading: There will be weekly written assignments, plus some class presentations. There will be no formal exams. Enrolled graduate students who have been admitted to candidacy will be excused from the written assignments.

Classes

Room: 004 Kemeny
Lectures: Monday-Wednesday-Friday 11:15 am--12:20 pm (11 hour)
X-hour: Tuesday 12:00 pm - 12:50pm

We may meet several of the x-hours, but this will always be announced in advance.

There will be no classes the week of October 24.

Staff

Instructor:
Carl Pomerance -- 339 Kemeny / Tel. 6-2635
Office hours: Tuesday and Thursday 9:00 AM--9:55 AM and by arrangement at other times.

Homework

Homework is due at the start of the class period on the due date.
Homework will be generally due once per week on Mondays.
Assignments will be posted on this website, with extra problems and/or comments added as the week progresses.

Past assignments

Homework due Wednesday, September 28:
Ch. 1, numbers 3,4,5,8,9. (If you see a problem you'd rather do, feel free to substitute.)
Also this problem: Let a1=1 and if an has already been defined, let an+1=an...a1+1. Prove that an<22n-1 for all n. Use this to show that π(x)≥loglogx for all x>1.

Here's a comment about Exercise 3. It is unexpectedly difficult and the referenced paper only proves a weaker result: Either n!+k has a prime factor larger than n or it is a power of k and k is prime. One then deduces the same, namely that among the n numbers n!+k for k running from 1 to n, there are at least n distinct primes dividing them.

The exercise as stated is true and it appears in a more difficult paper from 1979. I have worked out a proof of the original exercise along different lines and perhaps we can discuss it later in the term. For now, you are permitted to substitute the easier version discussed above.

Homework due Wednesday, October 5, 2011:
Try to do problems 13 and 22 in the book.
Use a computer to count the number of positive integers n with n prime and n at most 10k for k=2, 3, 4, and 5. Then do the same but instead of n being prime, count n if n2+1 is prime. Then, the same for n2+2. And then for n2+n+41.
Give another proof of Lemma 1.17 using smooth numbers.

Homework due Wednesday, October 12:
Homework problems

Homework due Wednesday, October 19:
Homework problems

Homework due Wednesday, November 2:
(1) Ch. 3, #1.
(2) Using the Taylor series for en, show that en > nn/n!.
(3) Ch. 3, #3. You can use problem 2 and also the second theorem of Mertens.
(4) Using results from this chapter, prove that the sum of 1/(p log p) as p runs over primes is convergent, but the sum of 1/(p loglog p) is divergent.
(5) Use the prime number theorem to show that there are infinitely many primes whose decimal representation start with the digits 123456789. (The first one is 1234567891.)
(6) Using the principle of partial summation, discussed in class, show that π(x) ~ x/log x as x tends to infinity if and only if θ(x) ~ x as x tends to infinity.

Exams

There will be no exams, but students will be expected to make some presentations to the class.