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Any suggestions for interesting problems?

  1. Dec 3, 2005 #1
    Hi everybody,
    I was wondering if anyone can suggest any really interesting math problem, as I seem to have reached a point where I am tired of reading theory and need to focus on some serious problem. I have studied Calculus 1 and 2(=single and multivariable calculus for Real numbers) ,some basic Linear Algebra, and some really basic Numerical Analysis. Currently, I work on elementary differential equations,an introduction to Probabilities theory and some basic ZF Set Theory. What I am actually looking for is not a difficult problem with a half-page solution but a difficult one with many questions that leads to interesting results.

    Thanks
     
  2. jcsd
  3. Dec 3, 2005 #2

    matt grime

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  4. Dec 3, 2005 #3
    How about the study of Groups of knots in [tex]R^n[/tex] n>3
     
    Last edited: Dec 3, 2005
  5. Dec 3, 2005 #4
    That's GREAT stuff matt!
     
  6. Dec 3, 2005 #5

    matt grime

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    There are mistakes in it, that i can see, but not terrible ones;. I wrote them from memory and didn't check somethings closely enough. I can write some more if needed, and make themlead onto the interesting things that the OP might want to consider (eg, prime decomposition in rings: what are the integers in Q[sqrt(d)] and all that, and algebraic number theory)
     
  7. Dec 3, 2005 #6

    benorin

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    This is seriously uncanny matt. I came accross this PDF on a google search not too long ago whilst looking up the principle of inclusion and exclusion (PIE) for my combinatorics HW. The last problem on that, I thought it was cool: so I posted it here.
    Wow, thanks matt.
     
  8. Dec 4, 2005 #7

    matt grime

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    Strictly speaking, most oft he answers to those questions are less than 1/2 a page, indeed are quite short, but may take a damn sight more than 1/2 a page to work out. Plus they were supposed to start you off towards graph theory, combinatorics, number theory, and so on, so if you find one you like i can develop the idea, or at least point you in the right direction to learn more. I suppose the most fruitful lead out would be number fields.
     
  9. Dec 4, 2005 #8
    Thanks for your replies and suggestions. I downloaded your problems matt and I think they may be what I am looking for. Although I am not very familiar with most of it, I think they are, as you say, "a good way to start you off towards graph theory, combinatorics, number theory". If I have any questions or ideas I will let you know.

    Thanks
     
  10. Dec 4, 2005 #9

    matt grime

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    Here's one thing for you to thinkabout.

    Although I think I've made some mistakes in the question about Z[sqrt(5)], what is the correct notion of prime and integer to allow us to generalize things and find integer solutions to equations. For instance, suppose we wanted to find all integer solutions to

    n^2=7p+9

    when p must be a prime?

    Rearrange and factor:

    (n-3)(n+3)=7p

    p and 7 are prime so it must be that

    n-3=7, n+3=p
    or
    n-3=p, n+3=7
    or
    n-3=1 n+3=7p
    or
    n-3=7p n+3=1

    which completely solves the problem.

    Now, what about finding all integer solutions to y^3=x^2+3?

    We can't factorize in integers, but we can factorize in the integers adjoing sqrt(-3) ie

    y^3=(x-sqrt(-3))(x+sqrt(-3))

    and if we knew about factoring now we could solve it.

    The problem is that we no longer know we have unique factorization. Indeed there are many false proofs of FLT that assume unique factorization (many believe one of these might have been the alleged proof too long for the margin).

    One quite well understood piece of mathematics is figuring out what primes and integers ought to mean in Q[sqrt(d)], so called number fields.

    You might think that the integers ought to be Z[sqrt(d)] inside Q[sqrt(d)], but you'd be wrong, or at least you'd realize that wasn't the correct generalization, algebraically. The correct definition of integer inside Q is that it is the root of a monic polynomial with coefficients in Z, thus in Q[srqrt(5)] we should consider phi, the golden ratio to be an integer since it satisfies a monic poly with integer coefficients. These are the socalled algebraic integers.
     
  11. Dec 4, 2005 #10

    Zurtex

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  12. Dec 4, 2005 #11

    matt grime

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    Does 212 imply this is the second year? The third question on the pint challenge shouldn't have the hint on it; the hint is the answer! The one about Q is good, but the second one can be done by citing a structure theorem, so presumably you aren't supposed to know the structure theorem of finitely generated abelain groups.
     
  13. Dec 4, 2005 #12

    Zurtex

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    Yeah this is 2nd year, but 1st year was very elementry, it mainly consisted of everyone catching up to people who had done Further Maths A Level.

    We have certainly not been taught any structure theorem in our course yet. As stated these were optional and with all the other stuff we had the 2nd one wasn’t designed to be overly hard. My proof for question 3 in the 2nd one was alas wrong but I rushed it in one night so I’m not too surprised. Other than that I got everything else from both papers right, the most interesting one I found to be the 2nd question on the 1st one.
     
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