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Find the sufficient condition for $l$ and $m$ for which the system of equations

$ab+bc+ca=l$
$abc=m$

has positive integeral solutions for $a,b,c$.

This is not a standard question from some source but I need for some work. I have found some conditions but they are not sufficient, but necessary.

Satvik Mashkaria
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  • Here's a sufficient condition: $$l=3 \qquad \text{and}\qquad m=1$$ It is, however, not necessary. – Zubin Mukerjee Jan 06 '15 at 15:17
  • @ZubinMukerjee Interesting, but what are the positive integer values of $a,b,c$ that works for $l=1$ and $m=1$? I would say $l=3$, $m=1$ is one set that gives results. – Martigan Jan 06 '15 at 15:20
  • @MarkBennet: That other question gives specific values for $l$ and $m$ and asks for all the integer solutions for those values. That makes this question quite different and a non-duplicate. – Rory Daulton Jan 06 '15 at 15:20
  • @RoryDaulton It gives those specific values as an example but asks for a general result. – Mark Bennet Jan 06 '15 at 15:21
  • @MarkBennet: That is not at all clear from that question, and in fact the only answer is only for those particular values of $l$ and $m$. And even if your interpretation is correct, that question asks for all integral solutions while this one asks only if there are any. – Rory Daulton Jan 06 '15 at 15:22
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    @RoryDaulton The two questions have been asked by the same person - if the first question is not clear enough then the remedy is to edit that question rather than to ask it again. – Mark Bennet Jan 06 '15 at 15:24

1 Answers1

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The formula I wrote, but it You don't like. Number of solution for $xy +yz + zx = N$

Then make the elementary transformations. Will present the equation is not much different form.

$$\left\{\begin{aligned}&ab+bc+ca=l\\&abc=m\end{aligned}\right.$$

Then the system can be rewritten in this form.

$$a=\frac{l-bc}{b+c}=\frac{m}{bc}$$

Means it is necessary to decompose the number $m$ on multipliers and to know when will be the identity. Because the number $a$ must be integer.

$$l=\frac{m(b+c)}{bc}+bc$$

I think this is an optimal search algorithm.

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