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I have this following problem:

$T=nP_1 + mP_2$

With $T$, $P_1$ and $P_2$ real numbers. I have access to those three values, but is it possible to determine $m$ and $n$ ? This looks like the Bézout identity but with real values instead of integers. So would it be a generalization? I also know that $m$ and $n$ are integers between $0$ and $8$.

For context, I am looking at the neighbors of a cell in an array, so 8 surrounding cells with 3 values possible and I want to retrive from the total sum of the values the linear combination of $P_1$ and $P_2$. So I don't have to use several loops and if statements slowing down my program.

Thanks in advance for your answer!

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If T, P1 and P2 are totally random, there is generally no solution with n and m integers. Even without the limintation between 0 and 8.

If they are not random, and combinations of 2 well-known values, you have a system with 2 equations and 2 values.

Ex :

$T = 5 \times \sqrt{2}+\sqrt{3}$
$P1 = 6 \times \sqrt{2}+4 \times \sqrt{3}$
$P2 = 7 \times \sqrt{2}-\sqrt{3}$

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  • $\begingroup$ Could you please specify the set of equations to solve because I can't put my finger on them ? $\endgroup$
    – Chris Ze Third
    Commented Oct 24, 2022 at 13:29
  • $\begingroup$ Also, $P_1$ and $P_2$ are set to $0.2$ and $0.3$ in my case. And T is necessarly a combination of them because I'm looking at the sum of neighbors of a cell in an array. Cell having a value of $0$, $P_1$ or $P_2$ $\endgroup$
    – Chris Ze Third
    Commented Oct 24, 2022 at 13:32
  • $\begingroup$ With numbers like this (decimal numbers), in many cases, you have many options. example : (n=3,m=0) and (n=0,m=2) give same result. So there is no direct method. Just try all values for n (9 values only) , and for each of them, check if m is a valid number. $\endgroup$
    – Lourrran
    Commented Oct 26, 2022 at 11:47

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