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please forgive me if this question is not a good one, I am just a high school student (note: this isn't homework).

I was wondering if every sequence of integers could be represented by an explicit equation.

I have read another similar question whose answer mentions a "Lagrange Interpolating Polynomial", but how would you find this polynomial? And what is the maximum length of sequence this works for?

Edit: I am specifically wondering how one would find an equation that fits, say 20, points, with the y values being randomly chosen integers. how would one go about finding this equation, and is this practical?

Thank you in advance for any answers.

cole-wilson
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  • Not sure what an "exit equation" is. Lagrange interpolation is used if you have $n+1$ points $(x_i,y_i)$ and you want to find a polynomial $a_0 + a_1 x +a_2 x^2 + ... + a_n x^n$ that passes through all of them. – Zadig Mar 27 '21 at 16:12
  • I assme you meant "explicit" equation, not "exit" equation? In any case, there are counting arguments involved here. In brief, there are only countably many "explicit equations", whatever that might mean, but there are uncountably many sequences. But if you are not familiar with arguments of that form, this may not be the ideal one to start with. – lulu Mar 27 '21 at 16:14
  • Yes I meant explicit, sorry! – cole-wilson Mar 27 '21 at 16:17
  • I am specifically wondering how one would find an equation that fits, say 20, points, with the y values being randomly chosen integers. how would one go about finding this equation, and is this practical? – cole-wilson Mar 27 '21 at 16:20
  • Lagrange Interpolation comes with an explicit construction. – lulu Mar 27 '21 at 16:23
  • Interpolation has many applications and is definitely useful. There are many ways you can do it (i.e. Lagrange interpolation, Newton interpolation). – Zadig Mar 27 '21 at 16:24
  • Thanks everyone! This was helpful. – cole-wilson Mar 27 '21 at 16:30

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