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For example, there is the Abel–Ruffini theorem that states there is no analytical solution to polynomial equations of degree five or higher, mentioning $x^5-x-1=0$ as the simplest example. Now if we take $$x^5-x-a=0$$ we can calculate $x$ numerically for every value of $a$ without any trouble. So clearly all the information necessary is contained in the equation. But despite this, we are not able to provide a formula like $x(a)=\dots$.

How is it possible that we have all the information necessary, but we still can not shape it into an explicit formula? What property is it that makes some equations solvable analytically, and others not?

MaxD
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    Oh, but you can provide a formula like $x(a) = \ldots.$ You just have to invent some parts of it. Tons of mathematics consists of functions that people invented in order to be able to solve problems. For example, $x = \sin(10^\circ)$ solves a problem that is just as "unsolvable" as $x^5-x-a=0.$ – David K Feb 11 '21 at 01:03
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    Considering we have only finitely many mathematical functions and constants we care to have given names and uncountably many real numbers, this is really no surprise. – K.defaoite Feb 11 '21 at 01:12
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    Asking for analytic solutions alone is very restrictive. So I guess by formula you mean a finite expression in the elementary function. Otherwise we can express every solution of such equations by a formula. In fact that is what a numerical solution is -- it is just a truncated form of an infinite expression representing the solution. – Allawonder Feb 11 '21 at 16:03

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What does it mean to solve a problem analytically? Finding "analytic" solutions to a cubic polynomial, I might have to write down numbers like $3^{1/3}$. What is $3^{1/3}$? Is that considered "analytic" to you? Radicals are just ways to write down a family of numbers not equal to the integers and not equal to a fraction of integers, just like how fractions are a way to write down numbers that are not equal to an integer. That theorem states that there are solutions to degree 5+ polynomials that cannot be stated in terms of integers, fractions, and radicals (read the first sentence of the wiki page). But like we did with radicals, we can invent a new way to write down this new class of numbers. I forgot what they are called but they are often written like $B(4)$, just like how $4^{1/4}$ is a thing. Writing down something "analytically" really depends on how you define analytic.

K.defaoite
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Spencer Kraisler
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  • Just found this cool table: https://en.wikipedia.org/wiki/Template:Mathematical_expressions. Guess what I call "analytical solution" is actually a closed-form expression. "Mathematical expression" seems to be the most general category, so could we come up with a mathematical expression like $x(a)=\dots$? Or do the symbols we would need just not exist (yet)? – MaxD Feb 11 '21 at 01:20
  • @MaxD No, they exist and are well understood. Check out algebraic numbers. – Spencer Kraisler Feb 11 '21 at 22:58
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You cannot obtain the analytical solution of the equation but you can obtain explicit asymptotic expressions of $x$.

To simplify the notations, let $a=b^5$ and we shall try to solve $$x^5-x-b^5=0$$ assuming $b>0$. Using high-order methods (order $n$) we shall have $$x_{0}=b+\frac{b}{5 b^4-1}$$ $$x_{1}=b+\frac{5 b^5-b}{25 b^8+1}$$ $$x_{2}=b+\frac{25 b^9+b}{125 b^{12}+25 b^8+5 b^4-1}$$ $$x_{3}=b+\frac{125 b^{13}+25 b^9+5 b^5-b}{625 b^{16}+250 b^{12}+50 b^8-5 b^4+1}$$ $$x_{4}=b+\frac{625 b^{17}+250 b^{13}+50 b^9-5 b^5+b}{3125 b^{20}+1875 b^{16}+500 b^{12}+6 b^4-1}$$ $$x_{5}=b+\frac{3125 b^{21}+1875 b^{17}+500 b^{13}+6 b^5-b}{15625 b^{24}+12500 b^{20}+4375 b^{16}+375 b^{12}+35 b^8-7 b^4+1}$$ Suppose that we do this forever.

Using $b=3$, some results

$$\left( \begin{array}{ccc} n & x_{n} & \text{decimal representation} \\ 0 & \frac{1215}{404} & 3.007425742574257425742574 \\ 1 & \frac{246645}{82013} & 3.007389072464121541706803 \\ 2 & \frac{100137870}{33297277} & 3.007389162783491274676905 \\ 3 & \frac{6254765325}{2079799117} & 3.007389162671713933610637 \\ 4 & \frac{33012650893518}{10977179575985} & 3.007389162671662103584418 \\ 5 & \frac{13403136063083787}{4456734841451944} & 3.007389162671662635777060 \\ 6 & \frac{680209145067512871}{226179289833989464} &3.007389162671662634787269 \\ 7 & \frac{2209319270264084987187}{734630322436023080017} & 3.007389162671662634787290 \\ 8 & \frac{896983610363648795857113}{298259906465446908872354} & 3.007389162671662634787294 \end{array} \right)$$

Claude Leibovici
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