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Math challenge

Hi guys,
This is my first time here so be gentle : )
I have this math riddle which is waaay over my league.
Maybe you could help out:

Let S be the following set of integers:
S = {3125   ,    46656    ,   65536   ,   823543  ,  16777216   ,  387420489}

What are all the solutions for the vector [a,b,n,c] for assigning the integers above (Si) in the following equation (I will break it down to parts I,J,K,L,M for simplification):

I = Si * b * ln(n)
J = a * sqrt( 1 + b^2 * ln(n)^2 )
K = a * ln(n)
L = ln( 1 + I / J )
M = L / K

c = cos(M) / sin(M)

any ideas?
P
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pixiepowder
Asked:
pixiepowder
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3 Solutions
 
d-glitchCommented:
What is the context of this question?
Where did you get it and why do you want to solve it?
Why do you believe it has a solution?

It doesn't appear to be homework.
But I'm afraid it doesn't really appear to be coherent either?

Maybe if you could scan the page that has the problem on it and post that...
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pixiepowderAuthor Commented:
hi dg,
It is no homework or any assignment for that matter, just a challenge.
I don't know if it has a solution or not, but proving there are no solutions is as good a solution as any.
What do you mean by not coherent?
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d-glitchCommented:
What does this mean      I = Si * b * ln(n)      ?

J K L and M appear to be numbers, if  I is a function of Si  it would seem to be a vector.

I don't see the relationship between S,  [a,b,n,c], and  I J K L M.


>> Where did you get it and why do you want to solve it?

Without some additional information and some context, by best guess is that this is
gibberish and/or a practical joke.


>> Maybe if you could scan the page that has the problem on it and post that...

If you think this is over your head, you shouldn't try to simplify it for us.  You might be
leaving out something important.  I would like to see the original and complete
statement of the problem.


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d-glitchCommented:
For what it's worth, I know where the large integers come from ...
S = { 3125,  46656,  65536,  823543,  16777216,  387420489}


5^5 =      3125
6^6 =     46656
7^7 =    823543 
8^8 =  16777216
9^9 = 387420489

Open in new window

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d-glitchCommented:
Sorry.  Missed one....
2^2^2^2 = 2^16 = 65536

Open in new window

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pixiepowderAuthor Commented:
oh, good. you made some sense of it already.
So it is not gibberish you see (not that I considered it was).

Here,
Let me see if I can make it easier for you to understand (and bear in mind I am not a mathematician):
The sole purpose of the I , J , K ,L , M  notations is for ease of writing, not to have to write all at once a very long and cumbersome equation that is equal to c .
Just place the M phrase in the last equation, the L and K phrases in the one before that, and so on.
The unknown constants are a , b , n , c and the variable is Si (which is members of the S set).

Maybe now, fueled by your discovery and this second attempt of explaining, it will be easier to address.

P
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d-glitchCommented:
I am quite sure this is gibberish.

This sentence makes no sense:

      What are all the solutions for the vector [a,b,n,c] for assigning the integers
      above (Si) in the following equation ...

What does it mean to assign an integer?

>> Maybe if you could scan the page that has the problem on it and post that...

And if you can't, I won't be posting again.
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pixiepowderAuthor Commented:
Goodbye sir,
Maybe someone else won't give up so fast.
Take care.

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SuperdaveCommented:
I take it Si means either "for some element of S", or "for each element of S", probably the latter.
The other thing that isn't stated although I wonder about it because of the type of problem and its inputs, is whether the answers are supposed to be restricted to integers?
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pixiepowderAuthor Commented:
hi there Superdave,

You're right, Si means "for each element of S".
The answers, meaning the values of a, b, n, c , are not restricted to integers. Real numbers (and even imaginary numbers) are also legitimate solutions.

P
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TommySzalapskiCommented:
It's a system of six equations and four unknowns. For simplicity, remember that cos/sin = cot. Also note that n never appears except as ln(n), so let's define ln(n) = d and solve for it remembering that d>0. These then are the six equations.
c = cot(ln(1+(3125*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
c = cot(ln(1+(46656*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
c = cot(ln(1+(65536*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
c = cot(ln(1+(823543*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
c = cot(ln(1+(16777216*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
c = cot(ln(1+(387420489*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))

What makes it complex is that there are 2 quadratics and the trig function. Since there are more equations than unknowns, it is quite possible that no solution exists.
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TommySzalapskiCommented:
I'm sorry. d can be anything of course. n > 0.
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pixiepowderAuthor Commented:
Thank you Tommy,
What you write makes perfect sense.
Can anybody here solve this manually or via programming?
P
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TommySzalapskiCommented:
I plan to when I get around to it (just had a new baby). If someone beats me to it, that's fine too. I tried Wolfram Alpha, but the problem was too long for it. Maybe it could be simplified to be short enough.
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pixiepowderAuthor Commented:
Major congrats Tommy!
Nothing beats that, no need to apologize.
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TommySzalapskiCommented:
Okay. I noticed some more that should help.
Obviously, the only thing that changes from one equation to the next is the Si, so if a solution exists it must be due to the fact that cot is periodic (repeats itself) with a period of pi (which simply means that cot(x) = cot(x+pi) = cot(x+2pi) = cot(x+y*pi) where y is any integer.
Let's analyze the equations a bit.
First, we need to remember that a and d can't be 0 since they are in denominators. Also, b can't be 0 since that would make c = cot(ln(1 +0)) = cot(0) = error since cot(0) = cos(0)/sin(0) = 1/0. So only c could be 0 if we wanted a 0 (which we might, who knows? 0 is easy to work with usually).
Now I want to see if a solution is possible:

c = cot(ln(1+(Si*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d))
tan(c) + y*pi = ln(1+(Si*b*d)/(a*(sqrt(1+b^2*d^2))))/(a*d)
e^(a*d*tan(c) + a*d*y*pi) = 1+(Si*b*d)/(a*(sqrt(1+b^2*d^2)))
e^(a*d*tan(c))*e^(a*d*y*pi) = 1+(Si*b*d)/(a*(sqrt(1+b^2*d^2)))
Let's define some new symbols to clean it up some
F = e^(a*d*tan(c))
G = a*d*pi
H = b*d/(a*(sqrt(1+b^2*d^2)))

Now we have
F*e^(G*y) = 1 + H*Si
where y is an integer.
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TommySzalapskiCommented:
To get y by itself, we can do
y = ln((1 + H*Si)/F)/G
I see no reason why there shouldn't be solutions.
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pixiepowderAuthor Commented:
Hi Tommy (and all the experts out there),
any idea how we can get a solution out of the insights mentioned above?
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pixiepowderAuthor Commented:
TommySzalapski was kind and bright enough to address my question, and although I didn't get it fully answered, I sure have advanced thanks to him.
Thanx Tommy!
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