1

It is known that if we only consider the first four decimal digits of the following (no rounding), we have that this holds:$$\pi^4+\pi^5=e^6\label1\tag1$$However, if we use even one more decimal place, we find that it does not, in fact, hold. So obviously, I want to find some $\epsilon$ (for $\epsilon\gt0$, since $\pi^4+\pi^5\lt e^6$) such that $\eqref1$ is true.

In other words, I want to find an $\epsilon$ such that$$(\pi+\epsilon)^4+(\pi+\epsilon)^5=e^6$$which means that I need to find the real root of the quintic$$x^5+x^4-e^6=0\qquad\boxed{\small{\text{Where }e\text{ is Euler's number}}}$$So I want to know if there is a way to solve quintics of the form$$ax^5+bx^4+c=0$$I specifically have in this case$$\boxed{a=1\\b=1\\c=-e^6}$$however if a general form of the quintic is known, that would be appreciated.

CrSb0001
  • 2,820
  • $x^5+x^4-e^6$ has exactly one real root, namely $x=3.14159268251$. So let $\epsilon=x-\pi$. There is no "formula" for a quintic, like for a quartic, cubic or quadratic. – Dietrich Burde Mar 19 '24 at 19:54
  • 2
    This might be helpful ? https://en.wikipedia.org/wiki/Bring_radical#Series_representation – Donald Splutterwit Mar 19 '24 at 19:56
  • 3
    https://en.wikipedia.org/wiki/Quintic_function#Solvable_quintics has a small collection of solvable quintics -- notice that most do not have variable coefficients. For your quintic, you could reduce to $x^5 + bx^4 - c = 0$ since one can just divide through by $a$. Your specific quintic has one real root $\epsilon = 2.892211149644086830{\dots} \times 10^{-8}$. – Eric Towers Mar 19 '24 at 20:02
  • Wolfram Alpha gives the closed form here with Glasser’s derivation, like on Wikipedia, likely being able to be used to derive it. – Тyma Gaidash Mar 20 '24 at 17:42

1 Answers1

2

Based on the theory explained here your problem can be written in terms of the Lambert-Tsallis function as:

$$x= \frac{W_{1/4}\bigg(e^{3/2}\cdot \frac{1}{4}\bigg)}{1/4}$$

with $ W_r = { +0.7854 + 0.0000i \\ +0.2089 - 0.7841i \\ +0.2089 + 0.7841i \\ -0.7266 - 0.4836i \\ -0.7266 + 0.4836i }$

and $ x ={ +3.1416 + 0.0000i\\ +0.8355 - 3.1366i\\ +0.8355 + 3.1366i\\ -2.9063 - 1.9343i\\ -2.9063 + 1.9343i}$

ZKZ
  • 582