A Simple Calculus view on Fermat's Last Theorem

Question

My question (more of a hypothesis really) is basically this: If a function $f(x)$ is defined such that $f'(x)$ is not constant and never the same for any 2 values of $x$. Then there do not exist positive integers $a,b,c$ and $a\le b\le c$ such that,

$$\int_0^{a} f(x)dx = \int_b^{c} f(x)dx\tag1$$

Taking $f(x)=x^n$ then

• for $n \gt 1$, $(1)$ becomes Fermat's Last Theorem, while
• for $n=1$, $(1)$ becomes $a^2 + b^2 = c^2$

Maybe this has something to do with the "curviness" (I am 16 so please forgive me for non-technical language) of the graphs of $x^n$ since for $n=1$ the graph is linear and for $n>1$ the graph is curvy.

My school teachers are not talking to me about this because "it is out of syllabus", so I thought maybe this community could help.

Answer 1

Interesting. But not true. If you define $f(x)=\dfrac1{x+1}$, then$$\int_0^1f(x)\,\mathrm dx=\int_1^3f(x)\,\mathrm dx,$$since$$\int_\alpha^\beta f(x)\,\mathrm dx=\log\left(\frac{\beta+1}{\alpha+1}\right).$$

Answer 2

(Too long for a comment, therefore adding as an answer)

Creating conjectures is a great way to challenge your understanding and improve your skills. Seeing that you're only 16 and you're already creating conjectures involving integrals that your teachers are avoiding to tackle, I thought I'd give my 2 cents (reminds me of myself).

As for your specific question, there are many counterexamples, as others pointed out. I'm going to talk about some general ideas of creating conjectures as a way to improve your skills.

What I'd like to say is basically that you should give a lot of thought in your choice of hypothesis. This is not silly nitpicking, this is a serious advice - good conjectures always have meaningful and logical hypothesis. By questioning yourself why you're using certain hypothesis often makes it clearer if your conjecture makes sense or not.

For example:

a function $f(x)$ is defined such that $f'(x)$ is not constant and never the same for any 2 values of $x$

I know you didn't say this, but assume for a moment that $f'$ is continuous. If this is the case, it is clear that from your hypothesis, $f'$ must be strictly monotonic. So, question yourself if you have good reasons to include in your conjecture functions that are differentiable but not $C^1$. If yes, why? (I don't see any reason). If not, then you should choose a less convoluted statement (such as "$f'$ is strictly monotonic").

(Also, note that "is not constant" is redundant anyway)

Also, I know it's cool that your conjecture becomes Fermat's Last Theorem in a certain case, but you should also question yourself what fundamental difference does it make that your integration limits are integers. Since your hypotheses are incredibly broad, I don't see why integer limits would have anything special - again, this is a suggestion for you to rethink your hypothesis. Why integers are special here? Why not allow any reals? By asking yourself this kind of question, you have the means to assess the value of your own conjecture yourself.

(Observe, for example, that if your conjecture was true, it would be trivial to prove a broader version for rational limits, by shrinking/enlarging $f$ by a constant...)

Keep up the good work!