Re: help with diophantine equation


En el mensaje:1124735187.920841.259920@xxxxxxxxxxxxxxxxxxxxxxxxxxxx,
john_ramsden@xxxxxxxxxxxxxx <john_ramsden@xxxxxxxxxxxxxx> escribió:
> mechmech wrote:
>>
>> I have this equation: 3*n*n = 3*k*k + 73*k + 14
>>
>> I am interested in the most efficient way to get the solution
>> ( which is n=32 and k=22 ) besides the trial and error method
>> ( k=1,2,3,,,, then calculate n ) because for big enough numbers
>> n and k, the trial and error is useless.
>
> It's a difference of squares, i.e. after multiplying every term
> by 12 it can be rearranged as:
>
> (6.k + 73)^2 - (6n)^2 = 5161
>
> which means the left hand side can be factored as follows:
>
> (6.(k - n) + 73).(6.(k + n) + 73) = 5161
>
> So the most efficient method of solution is to list every
> way in which 5161 can be expressed as the product of two
> factors (including both negative!):
>
> +- 1, +- 5161
> +- 13, +- 397
> +- 397, +- 13
> +- 5161, +- 1
>
> and equate linear factors to each pair and see if the
> result gives integers k and n. (This will be the case
> if any only if k - n and k + n are both integers of
> the same parity, i.e both odd or both even.)

Hi mechmech,

Apart from (n, k) = (32, 22), there is another solution, as you can see
following john_ramsden message.


--
Best regards,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosaQUITARMAYUSCULAS@xxxxxxxxxxx




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