Math 126: Theory of numbers

MacQuarrie Hall 235, MW noon-1:15pm (Sec. 01, code 42296)

Last update of this page: Thu Nov 05, 9:00pm

Announcements:

• Problem sessions for our class will be held every Fri, noon-1pm in MH424.

Administrative handouts:

Course notes:

Supplemental material: Alternate forms of the Fundamental Theorem of Arithmetic (Ch. 7).

Data: The first 500 primes, in text, Excel spreadsheet, and csv formats.

Number theory online: Indexed by related chapter in the text.

• Website for A Friendly Introduction to Number Theory, maintained by the author.
• (Ch. 12) The article Prime number races, which gives a more detailed examination of what happens with Dirichlet's Theorem on Primes in Arithmetic Progressions. Fun and accessible, with ideas for student research projects.
• (Ch. 13) The Twin Prime Conjecture Song (from the PBS show NOVA), on the 2005 work of SJSU's own Dan Goldston (with Janos Pintz and Cem Yildirim). You can also read a technical description here, though after the first 2 pages or so, it's only accessible to people with Ph.D.'s in math.
• (Ch. 13) In a manner typical of modern mathematics, the work of Goldston-Pintz-Yildirim led in an unexpected way (expect the unexpected!) to recent work by Ben Green (who spoke in the SJSU Math Colloquium last year) and Terence Tao proving that primes exist in arbitrarily long arithmetic progressions.
  • For a guide to the Green-Tao theorem, start with these slides (July 19, 2009) from Tao's blog.
  • Andrew Granville's paper on prime number patterns is another accessible account of the Green-Tao theorem and many wonderful consequences that follow from it.
  • For technical details, Tao's number theory page has pretty much everything you could ever want to know on this topic.
• (Ch. 12-13) The Prime Pages (references and resources about prime numbers).
• (Ch. 14) The Great Internet Mersenne Prime Search.
• (Ch. 14) Richard Guy's award-winning article The Strong Law of Small Numbers, on patterns that work for a while and then stop, and a few that go on forever. Many such patterns involve primes.
• (Ch. 19) We won't cover this chapter, but one notable and relatively recent development in primality testing is the AKS primality test, which is notable both because it is the first "practical" (polynomial running time) primality test that does not rely on randomness/guessing, and also because two of its co-creators were undergraduate students at the time. For a brief description of the test and the hoopla surrounding its discovery, see this article from the Notices of the AMS. In fact, you can read the paper itself here (it's surprisingly readable).

Outline answers/notes:

Topics for exams:

Sample exams:

Problem sets:

HW	Outline due	Due date	Last revision due	Problems
PS01	Mon Aug 31	Wed Sep 02	Mon Sep 13	1.1, 1.3, 1.4, 2.2, 2.6, 3.3.
PS02	Wed Sep 09	Mon Sep 14	Mon Oct 12	5.1, 5.4, 6.1(b), 6.2(b), 6.4, 7.2.
PS03	Wed Sep 23	Mon Sep 28	Mon Oct 12	See handout.
PS04	Wed Sep 30	Mon Oct 05	Mon Nov 16	See handout.
PS05	Wed Oct 07	Mon Oct 12	Mon Nov 16	11.3, 11.6, 11.10, 11.11(b,c), 12.2, 12.3(a,b), 13.5. (13.1 is no longer assigned.)
PS06	Wed Oct 21	Mon Oct 26	Mon Nov 16	See handout.
PS07	Wed Oct 28	Mon Nov 02	Mon Dec 07	16.1, 16.3, 16.5, 17.1, 17.3(a,c), 17.5(a,b).
PS08	Wed Nov 04	Mon Nov 09	Mon Dec 07	18.1, 18.2(a,c), 20.2, 20.3, 21.1, 21.2.
PS09	Mon Nov 16	Mon Nov 23	Mon Dec 07	21.6, 21.9 (experiment, conjecture, prove), 21.13 (experiment and conjecture only), 22.2, 22.3 (try many different values of p), 23.1.

Reading assigned in course:

Other useful links:

• If you're interested in typesetting math, the system I use to make my handouts is called LaTeX, and it can be downloaded for free (in Windows format) at http://www.miktex.org. If you own a Mac (system OS X) or a Linux computer, you already own a copy of this program, as it's been a standard Unix program for about 20 years.
  LaTeX is well worth learning, if you have some free time; it's used for manuals and technical reports throughout nerd-dom. For self-study, I highly recommend the LaTeX book by Leslie Lamport, which you can purchase here.
• Free downloads: Adobe Acrobat (for reading .pdf files)

• Instructor: Tim Hsu.
• Required text: A Friendly Introduction to Number Theory, Joseph H. Silverman, 3rd ed.
• Homework: Outline and final version due once each week, except in exam weeks. For details, see the handout on homework in Math 126.
• Exams: Three in-class 75-minute exams, plus 2 hour, 15 min final exam (Thu Dec 15, 9:45am-noon).
• Grading: Final course grades consist of:
  

  Homework:	20%
  Exam 1:	14%
  Exams 2 & 3:	18% each
  Final:	30%