Perhaps you’re referring to one of Utpal Banerjee’s books?

https://www.thriftbooks.com/a/utpal-banerjee/1265627/?srslti...

I owned a few of them along with Michael Wolfe’s book, Allen & Kennedy, etc when I was working in this space.

Hn upvotes the weirdest things. You're talking about polyhedral analysis which emphatically does not work on diophantine forms

http://polyhedral.info/

Probably in the future nobody will be able to figure out how these things are written, because studying math doesn't make as much money as vibe coding.

Don’t think that has anything to do with langlands..? Euclidean algorithm is literally just an explicit algorithm for prime factorisation

When saying “6 = 9 mod 3”, the “mod 3” is changing what “6” and “9” are referring to: they are now referring to the equivalence classes modulo 3 that 6 and 9 respectively belong to, and the statement expresses that those two equivalence classes are equal (hence using the equals sign is correct), are the same class. It’s a shorthand for “[6]_3 = [9]_3”, or whatever your notation is for equivalence classes.

I don't think quibbling over "equivalence" vs "equality" is useful personally. They're both "equality". Just what the sign "=" means differs depending on the type information.

E.g x: ZmodN == y: ZmodN is a different operation than x: Z == y: Z, but they're both the equality operation. We wouldn't makeup a new name for addition in this context.

It's also worth mentioning we can use the equality function (==): Z x Z -> bool to define the (==_N): ZmodN x ZmodN -> bool function in a semi-generic way. To do so, we "just" need a way to assign a unique representative (in Z) to any x: ZmodN. In other words, this is choosing a partial inverse to the reduction function modN: Z -> ZmodN. I think the partial inverse is a right inverse? so a function g : ZmodN -> Z such that (modN) o g x == x for all x in ZmodN. Anyway, given such a right inverse g, you can define (==_N): ZmodN x ZmodN -> bool via

(==_N) x y := (==) (g x) (g y)

This has the benefit that it doesn't treat ZmodN as special in any way. You can apply the same song and dance for more general quotient structures. This can be useful when doing e.g. matrix arithmetic, where you might want equivalence up to the choice of some rotation or something.

In Science I think this phenomenon is called consilience.

This is not bidirectional. The Davis-Putnam-Robinson-Matiyasevich theorem shows we can make a Diophantine equation that acts as a universal Turing machine, but there’s Diophantine equations that cannot be solved by Turing machines:

https://www.nlp-kyle.com/post/number_computability/

The smallest known Diophantine equation that cannot be solved by any Turing machine last I checked had ~8000 states as a Turing machine. This Turing machine cannot be decided to halt, and if it does halt in finite time then an (outer) Turing machine could execute it to predict that, so this lives beyond decidability:

https://scottaaronson.blog/?p=2725

I find it annoying that the response to this from the Chaitain perspective is to throw your hands in the air and say not all of math is predictable and let “equivalent to halting decidability” be the death of effort. There’s a richer field of ‘hypercomputation’ sitting beyond the pale, and I believe it will be topological applications that untwist this knot [pun intended]. I’m excited for the post Turing world but i dare say I won’t live to see it.

It is a surprising and delightful consequence of this observation that there is a polynomial whose integer-valued roots are exactly the set of all prime numbers.

you're thinking of vim

This is of course an oversimplification by graduate students trying to reach a broad audience, but one could argue that it in fact is. Langlands himself viewed his program as part of an attempt to achieve non-abelian reciprocity laws. See for instance: https://www.phys.ens.psl.eu/~kashani/slides_chenevier.pdf

They give a better example at the end of https://hidden-phenomena.com/articles/quadratic-residues

That's why 1 is not generally considered prime. If it was, 2 could be factored as 2, 2x1, 2x1x1, etc., which breaks the uniqueness property.