Number Theory
Field: Number Theory
We call a pair of prime numbers that have a difference of two “twin primes”.
Examples of twin primes include
(3, 5), (5, 7), (11, 13), (17, 19), . . . (2)
Are there an infinite number of twin prime pairs?
Writing this out in a mathy way. Define the set of twin primes to be
T = {(p, p + 2)|p, p + 2 are prime}. (3)
Is |T | infinite?
For this problem I would like you to write
- A background section detailing in simple terms the field and necessary information to understand your results
- A results section detailing in simple terms the prior proofs/experiments performed in the past
- A summary section detailing potential consequences of a proof of this problem, and how it could impact a sector of computer science
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These sections do not need to be long, as they are just so that I can sufficiently understand the following section. I do not have a minimum required length, however note that if you have each of these sections as one single sentence such that it appears you have only a surface level understanding on the material, I will grade accordingly. These three sections will be worth 50% of your final exam grade, and will be graded on how well you describe the material to me, as well as how well I believe you understand the material.
Next, I ask that you take the results and document the process of you “playing” with the results. Most research is gained through insight into other’s cut- ting edge research, and through new perspectives that the original researchers did not consider. Playing with the results can be, but are not limited to, any of the following
- Expanding on the results of a paper (better error bounds, considering different cases, )
- Applying new research to a problem
- Using your class knowledge to look at a problem in a new way
- Applying techniques from another field to a problem
Numerically computing large amounts of examples will not be good enough this semester. For example if you are working on Twin Primes you cannot ust calculate the first 100 twin primes.
This section will be also 50% of your final exam grade, and will consist of how well you articulate your thought process while attempting to expand on the results of the paper. You can get negative results (ie the stuff you came up with failed / is uninteresting) but so long as I can tell you made a clear attempt to attack the problem in a way you found interesting, and legitimately tried, I will consider that sufficient.
Number Theory
