We know the following equation:
xⁿ + yⁿ = zⁿ
Here, x, y, z, and n are all positive integers, and when n > 2, there exists no integer solution that satisfies this equation.
This is known as Fermat’s Last Theorem.
But this is not just a complex mathematical statement.
It’s not just about the equation — it’s about the structure.
There are no integer coordinates on a curve
Let’s try interpreting the equation structurally.
When n > 2, the equation xⁿ + yⁿ = zⁿ is no longer a straight line — it becomes a curve.
Points on this curve take the form:
(x, y) = (irrational number, irrational number)
In other words, the coordinates on the curve are composed of irrational functions.
This means that a point made of perfect integers
cannot exist on the curve at all.
The structure of the curve simply does not allow for precise integer mappings.
Straight lines can contain integer solutions
In contrast, when n = 1 or n = 2,
the equation describes a straight line.
In these cases, some integer coordinates can actually exist.
For example:
x² + y² = z²
This is the Pythagorean theorem.
A triangle like 3² + 4² = 5² exists with perfect integers.
But once n > 2, the equation becomes a curve again,
and integer coordinates can no longer be plotted.
The structure simply excludes integer solutions.
Structure excludes the answer
So we no longer need to treat this as
a “problem of finding integer solutions.”
The structure itself simply disallows them.
Fermat’s so-called “remarkable proof”
might have just been a moment of structural intuition.
“There are no points on the curve.”
That single sentence might have been enough.
Final Words
Mathematically, Andrew Wiles took hundreds of pages to prove it.
But in truth, this can be written down structurally in under three pages.
...But I’m busy, so I’ll go back to work now.
Several readers asked for a simpler overview instead of the full PDF.
This summary outlines the main principles of a volumetric 3D display based on multiphoton absorption in a low-energy plasma medium.
I've written a condensed summary version as a technical abstract here, in case anyone wants a quicker overview before diving into the full PDF:
https://blog.naver.com/as33sa/223833263822
This version focuses on the key principles, system architecture, and potential applications.
Since the original PDF is quite dense and spans 13 pages, I created this summary to make it easier for readers to grasp the core ideas more quickly.
Feedback is always appreciated!
Thanks for the comment! You're absolutely right that visual aids help a lot, especially for tech concepts.
Unfortunately, this is an early-stage patent-pending system, so I can't publish any diagrams or images that aren’t already protected. Also, since Hacker News doesn’t support inline images, it’s a bit tricky to show visuals here.
Still, the linked blog post provides a concept summary in English, and I’m happy to answer any technical questions if you're curious!
The concept has been theoretically validated, and the system is currently patent-pending.I’m now collaborating with an industry partner to develop a working prototype. Happy to go into technical details if you're curious!
I've read similar projects that use two lasers for microscopy [1]. They build them and used them to get images of cells. So I don't doubt of the general theoretical idea.
The devil is in the details. Can it run in open air or the concentration of argon is too low? Can it get enough brightness if you plug it in a standard electrical socket? How safe is it? [2] How many strong lasers escape the device? Does dust cause problems? Does it need a filter and a fan, or you need to enclose it to get clean enough air? How big it is? A tabletop device? Can it be shown to a thousand of people for an big event?
Perhaps you can get more traction here with some diagrams that show how it works without revealing all the secret details, but a text only explanation is too boring for the general public.
Anyway, I don't expect that this get many upvotes until you have a working prototype and some photos that shows it working.
[1] I think it was a German research team that also do confocal and 4pi microscopy. It was a long time ago, so I don't remember the details.
[2] A long time ago, I used an optical table that we shared with another experiment with a green 5W (5mW?) laser pointing to me and barrier in the middle and pipes and a lot of equipment. So it was safe, but just in case, never duck to pick stuff from the floor.
The system uses two intersecting CW lasers in ionized gas to induce spatially-localized visible light emission through multiphoton transitions (non-resonant, no intermediate state). It's designed for open-air operation and uses energy focusing via PWM and low duty cycles to stay Class 1-safe while maintaining sufficient photon density. We're developing a working prototype with an industry partner, aiming for tabletop-to-large-scale modularity.
Happy to go into more technical details if you have specific questions.
I've been developing a 3D display technology that uses multiphoton absorption in open air to generate visible light points in real 3D space — no screen or medium required.
Unlike laser plasma displays (which require femtosecond lasers and high power), my method achieves volumetric light emission using resonant multiphoton excitation, which dramatically reduces energy consumption and improves safety.
Potential applications include AR/XR displays, holographic signage, military targeting systems, and even immersive entertainment.
Here's the concept and basic theory, with technical sketches and explanation (in English):
https://blog.naver.com/as33sa
Would love to get any technical feedback, especially from folks working in optics, display tech, or photonics.
There are no integer coordinates on a curve Let’s try interpreting the equation structurally. When n > 2, the equation xⁿ + yⁿ = zⁿ is no longer a straight line — it becomes a curve. Points on this curve take the form: (x, y) = (irrational number, irrational number) In other words, the coordinates on the curve are composed of irrational functions. This means that a point made of perfect integers cannot exist on the curve at all. The structure of the curve simply does not allow for precise integer mappings.
Straight lines can contain integer solutions In contrast, when n = 1 or n = 2, the equation describes a straight line. In these cases, some integer coordinates can actually exist. For example: x² + y² = z² This is the Pythagorean theorem. A triangle like 3² + 4² = 5² exists with perfect integers. But once n > 2, the equation becomes a curve again, and integer coordinates can no longer be plotted. The structure simply excludes integer solutions.
Structure excludes the answer So we no longer need to treat this as a “problem of finding integer solutions.” The structure itself simply disallows them. Fermat’s so-called “remarkable proof” might have just been a moment of structural intuition. “There are no points on the curve.” That single sentence might have been enough.
Final Words Mathematically, Andrew Wiles took hundreds of pages to prove it. But in truth, this can be written down structurally in under three pages. ...But I’m busy, so I’ll go back to work now.