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Solving a formula


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#1 Joatmon

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Posted 14 October 2016 - 07:03 PM

I have reason to think that if A, B and N in the following equation are positive whole numbers then there is no solution. I would like either values of A, B and N which satisfy the equation or a reason why there is no solution. If you think it looks easy then give it a go - please.

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Edited by Joatmon, 14 October 2016 - 07:08 PM.

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Trying to make some sense of it all,

But I can see that it makes no sense at all


#2 imatfaal

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Posted 14 October 2016 - 09:20 PM

I am pretty sure not.  But cannot yet prove it.

 

If you rearrange for a  I think you get

 

a =\frac{ (\sqrt3 \sqrt{b (3 b^3+4 n^3)}-3 b^2+6 b n)}{(6 b)}

 

If b and n are +ve integers then it is obvious that every term except

 

 \sqrt3 \sqrt{b (3 b^3+4 n^3)}

 

is an integer.  So for a to be an integer then that term must also be an integer (ie if it is less than 1 then no matter how much you add to it - or whatever integer you divide it by it will always be less than 1. or if it is greater than 1 then it cannot be irrational for similar reasons).  I feel I should be able to show that cannot be the case - but I cannot


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#3 Joatmon

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Posted 15 October 2016 - 10:50 AM

I'm not surprised - if I have worked things out correctly then a whole number solution would lead to a whole number example of A cubed plus B cubed = C cubed. If it can be proved that there is no whole number solution then it can be concluded that there is no whole number solution to A cubed plus B cubed = C cubed.


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Trying to make some sense of it all,

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#4 imatfaal

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Posted 15 October 2016 - 09:59 PM

Well then there is your proof


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there shallow draughts intoxicate the brain, and drinking largely sobers us again.

- Alexander Pope

 

feel free to click the green arrow  ---->

 


#5 Joatmon

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Posted 16 October 2016 - 09:19 AM

Well then there is your proof

I think that sometimes a journey and its byways can be as interesting as a destination never reached. Also, I feel certain that there are a number of people in the world, such as me, who although knowing  they are almost certainly deluded, think that there is just a possibility that there is still to be found a much more simple way of proving Fermat's Last Theorem than that produced by Andrew Wiles'. :D


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#6 renerpho

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Posted 16 October 2016 - 06:32 PM

I think that sometimes a journey and its byways can be as interesting as a destination never reached. Also, I feel certain that there are a number of people in the world, such as me, who although knowing  they are almost certainly deluded, think that there is just a possibility that there is still to be found a much more simple way of proving Fermat's Last Theorem than that produced by Andrew Wiles'. :D

 

Perhaps - but notice that, even if your formula is related to Fermat's Last Theorem for the case n=3, it can not attack the general case. And there already are easy and elementary proofs for the case n=3.


Edited by renerpho, 16 October 2016 - 06:32 PM.

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#7 Joatmon

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Posted 16 October 2016 - 09:13 PM

That is true and also for other specific values of n. However, perhaps the source of the formula I gave might lead to a generalisation for all values of n. I admit that is unlikely but it is an interesting conundrum. 


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#8 imatfaal

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Posted 17 October 2016 - 10:42 AM

That is true and also for other specific values of n. However, perhaps the source of the formula I gave might lead to a generalisation for all values of n. I admit that is unlikely but it is an interesting conundrum. 

 

Absence of evidence is not absolute evidence of absence; but when the greatest mathematicians search fruitlessly for a simple solution for centuries I think that - on the balance of probabilities - one can assume that a simple solution does not exist.


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A little learning is a dangerous thing; drink deep, or taste not the Pierian spring:
there shallow draughts intoxicate the brain, and drinking largely sobers us again.

- Alexander Pope

 

feel free to click the green arrow  ---->

 


#9 Joatmon

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Posted 17 October 2016 - 04:10 PM

 

Absence of evidence is not absolute evidence of absence; but when the greatest mathematicians search fruitlessly for a simple solution for centuries I think that - on the balance of probabilities - one can assume that a simple solution does not exist.

Can't argue with that :)

I guess that if I got a chance to play chess against a grand master I would jump at the chance and come back for more after losing. I suppose its a question of what one finds interesting or fun :D


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Trying to make some sense of it all,

But I can see that it makes no sense at all





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