u^3 - u + v^3 - v = w^3 - w

Discussion in 'Physics & Math' started by itchy, Jul 10, 2002.

  1. itchy Registered Senior Member

    Messages:
    47
    u,v,w are Z+

    Perhaps someone brainy can recognize the above equation. I've tried to find a general pattern to ALL the solutions but have failed so far. It's not an easy pattern.

    Can anyone give me an infinite number of solutions? If not, then whats the largest solution you can find? (largest w)
     
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  3. BloodSuckingGerbile Master of Puppets Registered Senior Member

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    I don't know the pattern of the roots of the equation but if you write the equation like this:

    u(u^2 - 1) + v(v^2 - 1) = w(w^2 - 1)

    you get

    (u - 1)u(u + 1) + (v - 1)v(v + 1) = (w - 1)w(w + 1)

    (x-1)x(x+1) is the product of 3 following numbers, so you can write that as

    [(u+1)!]/[(u-2)!] + [(v+1)!]/[(v-2)!] = [(w+1)!]/[(w-2)!]

    and if you multiply everything by 1/(3!)
    you'll get

    [(u+1)!]/[3!*(u-2)!] + [(v+1)!]/[3!*(v-2)!] = [(w+1)!]/[3!*(w-2)!]

    and this equals

    _3______3______3
    C____+ C____= C
    _u+1____v+1____w+1

    (binomial coefficients)
    (Ignore the underlines. I used them coz the spaces get trimmed)

    so u^3-u+v^3-v=w^3-w becomes the above equation.

    So the equation has an infinite number of solutions.
    I guess you've probably figured that out...
    Hope it helps...
     
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  5. itchy Registered Senior Member

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    How do you know it has infinite number of solutions?
     
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  7. Merlijn curious cat Registered Senior Member

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    Gerbile,
    looks nice, well done, but this:
    "if you write the equation like this:

    u(u^2 - 1) + v(v^2 - 1) = w(w^2 - 1)

    you get

    (u - 1)u(u + 1) + (v - 1)v(v + 1) = (w - 1)w(w + 1)

    (x-1)x(x+1) is the product of 3 following numbers"
    seems enough to show there are an infinite number of solutions, becuase it is p + q = r, with p=(u - 1)u(u + 1), etc.
     
  8. overdoze human Registered Senior Member

    Messages:
    310
    Merlijn,

    it may be p+q=r, but with a stipulation that each of p, q and r is factorable into three consecutive integers...? How do you guarantee that r can be arbitrarily large? This sort of smells like finding the largest prime, which is an unsolved problem.
     
  9. Merlijn curious cat Registered Senior Member

    Messages:
    1,014
    the mmoment I posted I realised that it was a bit too thin! Well, I'll just bame this being pre-coffee time of the day.

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    but with regard to the prime thing:
    y = x!+1
    1< z <= x

    y/z = (x!+1)/z = x!/z + 1/z = [(z-1)!z(x!/z!)]/z + 1/z = (z-1)! (x!/z!) + 1/z.

    (z-1)!(x!/z!) is always element of N
    this means that y = x!+1 must be a prime, or can only be devided by numbers larger than x, making those numbers primes lager than x.
    x=5; 5!+1 = 121 is not a prime , but can devided by 11 (11>5 and is a prime)
    or x=3; 3!+1=7, 7 is a prime.

    Since x can be any number, there is no largest prime.
     
  10. overdoze human Registered Senior Member

    Messages:
    310
    Ooooh, that was beautiful. Thanks, Merlijn! (I must've been thinking of some other unsolved problem....)

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  11. Merlijn curious cat Registered Senior Member

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    1,014
    "Ooooh, that was beautiful. "
    thanks.... I must admit, I cried when I first wrote down the proof.
    I once read (in Goedel, Escher, Bach) that Euclides had proven there are an infinite amount of primes. All Hofstadter said that it came down to the x!+1. If you have that hint, it's quite easy to make up the rest, still the sheer elegant beauty of the proof overwhelemed me for a second.
     
  12. BloodSuckingGerbile Master of Puppets Registered Senior Member

    Messages:
    440
    Merlijn, thanks for the remark. I didn't notice that.

    I've noticed something that might help in finding the pattern of the answers.
    The product of three consecutive integers (thanks for widening my vocabulary, Overdoze

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    ) can be also represented using a sum of factorials.
    for example:
    u=4=>3*4*5=60=4!*2+3!*2
    u=5=>4*5*6=210=5!+3*4!+3*3!
    ...

    It's pretty obvious considering the relation of the equation to the binomial coefficients.

    Just in case you didn't notice.
     
  13. overdoze human Registered Senior Member

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    310
    Glad to be of service.

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    But...what the heck are you talking about??

    5! alone is 2*3*4*5 = 4*5*6, which is by the way 120 not 210

    :bugeye:

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  14. BloodSuckingGerbile Master of Puppets Registered Senior Member

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    sorry, I meant
    u=6=>5*6*7=210=5!+3*4!+3*3!
     
  15. overdoze human Registered Senior Member

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    310
    Ok, sorry for being so dense.

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    And I apologize again, but after hours of contemplation I don't see what you're describing. i.e. I don't see a pattern of factorial sums for 3 consecutive integer products. Could you please post a general formula you're using, or even better yet a derivation?

    As an aside:

    x^n + y^n = z^n

    is Fermat's last theorem, which says there are no integer solutions for n > 2. Though I'm not sure how that bears, if at all, on the particular form presented in this thread...

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  16. BloodSuckingGerbile Master of Puppets Registered Senior Member

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    I didn't find a pattern either, I just noticed it. I didn't find a formula for it or anything, however I'll try to derive it.
    I hope the derivation is not as long as for Fermat's theorem...
    by the way, why did you post it (Fermat's theorem)?

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  17. BloodSuckingGerbile Master of Puppets Registered Senior Member

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    I didn't find the pattern of the factorials, however I did notice that the integers formed by the product of three consecutive numbers

    (u-1)u(u+1)

    can be written as

    3!*C<sub>n</sub>

    where n=u-1 (u>1)

    It's obvious, because the product is equal to

    (u+1)!/(u-2)!

    and if you divide that by 3!, it will form a binomial coefficient, which is always N.

    So C<sub>n</sub>n is a series where

    C<sub>n+1</sub>=C<sub>n</sub>+b<sub>n</sub>

    and b<sub>n+1</sub>=b<sub>n</sub>+n+2

    b<sub>1</sub>=3.
    c<sub>1</sub>=1.

    I doubt that this would help in finding the pattern of all the solutions, but.. who knows?
     
  18. overdoze human Registered Senior Member

    Messages:
    310
    The reason for my mentioning Fermat, is that it is very similar in form to the problem posed here. Fermat says that

    u^3 + v^3 = w^3

    has no Z+ solution sets. Rewriting the above equation, you have

    u*u*u + v*v*v = w*w*w

    In the problem posed in this thread, you have

    (u-1)*u*(u+1) + (v-1)*v*(v+1) = (w-1)*w*(w+1)

    The forms are eerily similar. Which made me wonder if the equation has any Z+ solutions at all, let alone an infinite number of them. I don't know what I was thinking, since obviously there *are* solutions. e.g. w=5, u=v=4

    [edit: removed some raving lunacy I've included previously. Must've been something I ate...]
     
    Last edited: Jul 13, 2002
  19. BloodSuckingGerbile Master of Puppets Registered Senior Member

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    440
    Itchy, I think Overdoze was right and we were all wrong. The series I was talking about is really weird and I doubt that the sum of any C<sub>u</sub>
    with any other C<sub>v</sub> will be equal C<sub>w</sub>.
    I'll try to prove that and then I'll tell you for sure.

    Not everything is as obvious as it seems...
     
  20. BloodSuckingGerbile Master of Puppets Registered Senior Member

    Messages:
    440
    .......
    Today I was really bored so I derived the formula for C<sub>n</sub>.

    C<sub>n</sub>=(n<sup>3</sup>+3n<sup>2</sup>-16n+12)/6 +3n-2.

    C<sub>1</sub>=1
    C<sub>2</sub>=4
    C<sub>3</sub>=10
    C<sub>4</sub>=20
    C<sub>5</sub>=35
    ...

    And then I wrote
    C<sub>n</sub>+C<sub>n+k</sub>
    in order to prove that a sum of two terms of the series is not equal another term in the same series.
    That is eventually equal
    6(2C<sub>n</sub>+C<sub>k</sub>)+3nk(2+n+k).
    But that doesn't prove it ...or does it?

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    Work with me people!!
     

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