Firstly, there are three administrative notes.
- I've posted the first homework set. This is due on Thursday, and you can find it here.
- I haven't set official office hour times yet. But I will have office hours on Monday from noon to 2pm on Monday, 1 Feb 2016, in my office at the Science Library.
- If you haven't yet, I encourage you to read the syllabus.
We mentioned several good and interesting "number theoretic" problems in class today. I'd like to remind you of some of them, and link you to some additional places for information.
Pythagorean Theorem
We've found all primitive Pythagorean triples in integers, which is a very nice theorem for an hour. But I also mentioned some of the history of the Pythagorean Theorem and the significance of numbers and number theory to the Greeks.
I told the class a story about how the Pythagorean student who revealed that there were irrational numbers was stoned. This is apocryphal. In fact, there is little exact record, but his name was Hippasus and it is more likely that he was drowned for releasing this information.
For this and other reasons, the Pythagorean school of thought split into two sects, one from Pythagoras and one from Hippasus.
Goldbach's Conjecture
Is it the case that every even integer is the sum of two primes? We think so. But we do not know.
I mentioned the Ternary Goldbach Conjecture, also known as the Weak Goldbach Conjecture, which says that every odd integer greater than
Fermat's Last Theorem
Are there nontrivial integer solutions to
This is one of the most storied and studied problems in mathematics. I think this has to do with how simple the statement looks. Further, we managed to fully classify all solutions when
If time and interest permits, we will return to this topic at the end of the course. There is no way that we could present a proof, or even fully motivate the proof. But we might be able to say a few words about how progress towards the theorem spurred and created mathematics, and maybe we can give a hint of the breadth of the ideas used to finally produce a proof.
Twin Prime Conjecture
Are there infinitely many primes
This culminated with the Polymath8 Project Bounded Gaps Between Primes. Math can be a social sport, and the polymath projects are massively collaborative online and open projects towards math problems. They're still a bit new, and a bit experimental. But Polymath8 is certainly extremely successful.
What is known is that there exists at least one even number
The ideas that led to this result can likely be sharpened to give better results, but actually proving that there are infinitely many twin primes is almost certainly going to require a brand new idea and methodology.
The best related result comes from Chinese mathematician Chen Jingrun, who proved that every sufficiently large even integer can be written either as a sum of two primes, or as a sum of a prime and a number with exactly two prime factors. Although this seems very close, it is also likely that this idea cannot be sharpened further.
Writing Numbers as Sums of Squares, Cubes, and So On
Can every integer be written as the sum of three squares? What about four squares? More generally, is there a number
Similarly, is there a number
These problems are all associated to something called Waring's Problem, about which much is known and much is unknown.
We also asked which primes can be written as a sum of two squares. Although we might have a hard time finding those primes right now, you might try to show that if
Max's Conjecture
For primitive Pythagorean triples
We didn't return to this in class, but we can now. First, note that since
Max conjectured that it is always the case that
Writing
which can be rewritten as
So Max's Conjecture is true, and every number appearing as
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Comments (2)
2018-06-09 Trm2slo
Is the right side of eq (2) correct? Should the3 be a 9?
2018-06-09 davidlowryduda
Yes, you're right. Thank you for pointing that out. I've now edited it.