Linear independence of the numbers $\{1,e,e^2,e^3\}$

Since I've spent enough time thinking about this, yet not getting a proof, I might as well show what I've got. Others can comment on whether or not more can be done.

Your problem is solved if you can show that for any integers $a, b, c$, we have $$\sum^\infty_{n=0} \frac{1}{n!} (a + b 2^n + c 3^n)$$ irrational (using taylor series).

WLOG, assume $c>0$. Pick $N$ large so that $(a+b2^n+c3^n) > 0$ for all $N \geq 0$. Then our problem is equivalent to showing that the series with strictly positive terms $$\sum^\infty_{n=N} \frac{1}{n!} (a + b 2^n + c 3^n)$$ is irrational. Suppose it was not and equal to $p/q$. Now we try to mimic the proof of irrationality of $e$.

Define

$$x = q!\left(\frac{p}{q} - \sum_{n=N}^{q} \frac{1}{n!} (a + b 2^n + c 3^n) \right).$$ One easily sees by distributing that it is an integer, and because our original series contains only positive terms, $x>0$.

Note that we can also write $$x = \sum_{n=q+1}^\infty \frac{q!}{n!} (a + b 2^n + c 3^n).$$

Now if $b=c=0$, then using $q!/n! < 1/(q+1)^{n-q}$ gives a geometric series bound that gives $x < 1/q$. Then we can get $x<1$ which is a contradiction that $x$ is an integer.

The terms $2^n$ and $3^n$ grow too fast for this same trick to work. You'd get bounds of $2^q/q$ and $3^q/q$ respectively. Since $q!/n! < 1/(q+1)^(n-q)$ is not tight, it is still possible that we can get our sum under 1. Or maybe we can monkey with our original definition for $x$.

I think what really needs to be copied are proofs of the irrationality of $e^2$ and $e^3$, but I am not aware of such proofs. Googling, I found a very algebraic proof of the irrationality of $e^2$, but I didn't read it carefully. This suggests proofs of the irrationality of $e^2$ may not easily generalize, and hence you aren't really proving that $e$ is transcendental at the same time.

Below is my attempt which is too long for a comment and may be saveable, (doubt it).

Consider the differential equation $y^{(4)}-6y^{(3)}+11y''-6y'=\textbf 0$, where $\bf 0$ is the null function over some non-trivial interval $I$ containing $1$.

The theory of ODE tells us that a basis of solutions is $$\{\underbrace{x\mapsto 1}_{\large \varphi_0}, \underbrace{x\mapsto e^x}_{\large \varphi _1}, \underbrace{x\mapsto e^{2x}}_{\large \varphi _2}, \underbrace{x\mapsto e^{3x}}_{\large \varphi _3}\}$$

This implies that $$(\forall \lambda _0,\lambda _1, \lambda _2, \lambda _3\in \Bbb Q)\left[(\forall x\in I)\left(\sum \limits_{k=0}^3\lambda_k\varphi_k(x)=0\right)\implies \lambda _0=\lambda _1=\lambda _2=\lambda _3=0\right] \tag {*}$$

Now if we could somehow prove that $(*)$ would also hold for the intersection of all such intervals $I$ (containing $1$), what we want would follow. But I have no hope of this being doable.

I thought to add an answer instead of giving long comments.

From Wikipedia we have the following quote

"In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that $e$ is not a root of a third degree polynomial with rational coefficients. In particular, $e^{3}$ is irrational."

The reference quoted is Hurwitz, Adolf (1933) [1891]. "Über die Kettenbruchentwicklung der Zahl $e$".

Luckily after much searching I was able to find this reference in an old journal available on internet archive. Here Hurwitz analyzes the simple continued fractions of numbers related with $e$ and notices that most of them have terms in an arithmetic progression (after a certain point).

He then proves the following theorem:

If simple continued fractions of two positive numbers $x, y$ have terms which are in arithmetic progression (after a certain point) then we can't have a non-trivial bi-linear relation of the form $$y = \frac{Ax + B}{Cx + D}$$ with integer coefficients $A, B, C, D$ unless the terms in their continued fraction belong to the same arithmetic progression.

Then Hurwitz notes that $$x = \frac{e - 1}{2} = [0, 1, 6, 10, 14, 18,\ldots]$$ and $$y = \frac{e^{2} - 1}{2} = [3, 5, 7, 9, \ldots]$$ where notation $$[a_{0}, a_{1}, a_{2}, \ldots]$$ represents the continued fraction $$a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}}$$ And clearly both of them have terms belonging to arithmetic progressions ($6, 10, 14, \ldots$ and $3, 5, 7, 9, \ldots$ respectively) but these are not the terms belonging to same AP and hence there is no non-trivial bi-linear relation of type $$y = \frac{Ax + B}{Cx + D}$$ with integer coefficients $A, B, C, D$.

Now it follows easily that $1, e, e^{2}, e^{3}$ are linearly independent over $\mathbb{Q}$. If it was not the case then we have integers $a, b, c, d$ not all $0$ such that $$ae^{3} + be^{2} + ce + d = 0$$ Using $e^{2} = 2y + 1$ and $e = 2x + 1$ we get $$a(2x + 1)(2y + 1) + b(2y + 1) + c(2x + 1) + d = 0$$ which leads to $$Axy + Bx + Cy + D = 0$$ with $A, B, C, D$ as integers or $$y = -\frac{Bx + D}{Ax + C}$$ and this is not allowed by the theorem of Hurwitz mentioned above.

Unfortunately I could not understand the proof of his theorem on continued fractions (because the whole paper/journal is in German). With reasonable effort and Google Translate I was able to understand the gist of the paper and I have presented the same in this answer. I have asked for the proof of Huzwitz theorem on MSE.