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September 1st, 2017, 03:12 PM   #1
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Repetition in an Equation

Hi,

Could anyone please tell me why it is that the numbers returned from this equation, repeat (but upon the repeat they are 65535-x). Start with x=1 then keep looping this:

x=(75*(x+1))-INT ((75*(x+1))/65537)*65537-1

That is: x=MOD(75*(x+1),65537)-1

Upon 32768 iterations of this equation the next number will again start the sequence, but it will be of the form 65535-x=x(1)

What I'm trying to say is that 65535-x(32769)=x(1). I hope I am making it clear that 65535 minus the 32769th iteration is the same as the first iteration.

I expected that the numbers would repeat after 65536 iterations, but not half way through.

Any ideas on why it happens?

Thanks for any help guys,

matalog.

Last edited by skipjack; September 2nd, 2017 at 01:54 AM.
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September 1st, 2017, 04:17 PM   #2
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Wolfram|Alpha: Computational Knowledge Engine)
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September 1st, 2017, 11:07 PM   #3
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Thanks for replying, but I'm not exactly sure how that W|A link you have provided helps to answer my question.

Could you explain a bit more please?

Thanks,

matalog

Last edited by skipjack; September 2nd, 2017 at 01:44 AM.
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September 2nd, 2017, 03:14 AM   #4
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Let m denote your constant, 75, and s denote your starting value (of x), 1.

Let p denote the minimum number of iterations that results in x becoming 65535 - s.

It's easy to show that further iterations continue in that manner (so that the sum of the results of the 2nd and (p+1)th iterations is 65535, and so on).

I think it would be fairly easy to show that p is a function of m alone, which means that p divides 65536. It's easy to find values of m for which p < 32768. I haven't found a value of m for which p = 65536.
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September 2nd, 2017, 08:13 AM   #5
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Yes, a lot of modulos like this behave in a similar way.

I would like to know though, why they repeat 65535-x in the second half?

They seem to do an interesting thing, each iteration of this particular equation, returns an original number between 1 and 65536, between x(1) to x(65536), and half way through that, it uses the very same pattern, albeit 65535-x(32769)=x(1).

It seems special in some way, can anyone explain it to me in simple terms, why I should not be surprised by what this equation (and others like it) does?

Thanks again, for any help,

matalog.
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