See https://www.mdpi.com/2227-7390/10/23/4471

On the Nature of Some Euler’s Double Equations Equivalent to Fermat’s Last Theorem

In this work, I provide a new rephrasing of Fermat’s Last Theorem, based on an earlier work by Euler on the ternary quadratic forms. Effectively, Fermat’s Last Theorem can be derived from an appropriate use of the concordant forms of Euler and from an equivalent ternary quadratic homogeneous Diophantine equation able to accommodate a solution of Fermat’s extraordinary equation. Following a similar and almost identical approach to that of A. Wiles, I tried to translate the link between Euler’s double equations (concordant/discordant forms) and Fermat’s Last Theorem into a possible reformulation of the Fermat Theorem. More precisely, through the aid of a Diophantine equation of second degree, homogeneous and ternary, solved not directly, but as a consequence of the resolution of the double Euler equations that originated it, I was able to obtain the following result: the intersection of the infinite solutions of Euler’s double equations gives rise to an empty set and this only by exploiting a well-known Legendre Theorem, which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. The impossibility of solving the second degree Diophantine equation thus obtained is possible using well-known techniques at the end of 18th century (see Euler, Lagrange and Legendre) and perhaps present in Fermat’s brilliant mind.

@AndreaOssicini "The impossibility of solving the second degree Diophantine equation thus obtained is possible using well-known techniques at the end of 18th century (see Euler, Lagrange and Legendre) and perhaps present in Fermat’s brilliant mind."

19th century is two centuries too late, but I'll try to look into this.

@makoyass

see also https://arxiv.org/abs/1704.06335

I made a version that resolves in 2027 so those of us with short AI timelines who want to believe re Fermat are forced to actually risk our mana

@Lorec Note my one can resolve yes or no early when we get compelling enough AI mathematicians, which will might happen like... next year lol.

@makoyass I did see that! I just didn't want to be free from consequences in the case that, eg, AI mathematicians remain uncompelling for 6 years and then the world suddenly explodes.

Why would it make the world a dimmer place?

@Lorec It would mean that exhaustively searching mathematics is much harder, which means that there's a lot out there that we cannot see.

You could take that as good news that there's more to be discovered than we thought, but for me the main thing that stands out is that it tells me is that it's going to be much harder to understand and cultivate trust with other advanced beings, because they're going to be onto a lot of mathematics that we're not familiar with. I currently take solace in the fact that there only seems to be one convergent cooperative bargaining theory, which is a logic of multiversal peace. If math is harder to exhaustively explore than I thought, that may no longer be the case, and peaceful trade with non-human intelligences becomes a lot harder, and war threatens.

@makoyass If each of two civilizations has some theorems that seem to suggest war when they could have just negotiated, is there any reason they shouldn't be able to just put the theorems aside and negotiate, even if they won't be able to do it very formally?

@Lorec I guess my load-bearing assumption here [which is intuitive to me but may not be to others] is that civilizations should be able to mutually read each other as potentially cooperative on a level that doesn't rely on sharing the same formal math.