PRACTICAL MAGIC

[Gian]

I believe it was in 1995 that Andrew Wiles proved Fermat's Conjecture, that no integer greater than n=2 could possibly satisfy the equation aⁿ + bⁿ = cⁿ

Not knowing anything about Maths beyond GCSE I hope I've got that right.

Does Andrew Wiles' proof have any practical application? Eg does it allow space engineers or architects or anyone else to do something they could not do previously?

Or is this purely knowledge for its own sake? Not that there's anything wrong with that of course!

Either way it's still beautiful and fascinating

cheerz

GIAN🙂XXX

Edited July 4 by Gian

[joigus]

16 minutes ago, Gian said:

that no integer greater than n=2 could possibly satisfy the equation aⁿ + bⁿ = cⁿ 

Close enough.

that no integers a, b, c > 0 could possibly satisfy the equation aⁿ+bⁿ=cⁿ for n also integer and greater than 2. Otherwise there trivially are such a's b's and c's aplenty.*

I know of no practical applications of it. But I wouldn't rule them out.

* Counterexample: n=3, a=2, b=1, c=9^1/3

Edited July 4 by joigus
addition of counterexample+small correction

[MigL]

No practical application, maybe.

But it does provide a 'bound' on what can be done.
If you can't do it mathematically, you can't do it physically either.
( although the opposite is not always true; some things you can do mathematically cannot be done physically )

[Gian]

38 minutes ago, MigL said:

No practical application, maybe.

But it does provide a 'bound' on what can be done.
If you can't do it mathematically, you can't do it physically either.
( although the opposite is not always true; some things you can do mathematically cannot be done physically )

Agreed. I was interested to read in a kids' science book (my scientific level) that although there's no such number as √-1 but it's still possible to use it. They were able to deploy √-1 to prove faster-than-light travel would be impossible.

Gobsmacked!

Cheerz

GIAN😀

 

[Ghideon]

3 hours ago, joigus said:

I know of no practical applications of it.

That is, as far as I know, correct. There are some indirect applications in for instance ECC (Elliptic Curve Cryptography) and Elliptic Curve Digital Signature Algorithm (ECDSA). This may be of interest to OP since it is used, maybe on a daily basis, when browsing the internet; Elliptic Curve Cryptography (ECC) is of interest because it offers the same level of security with (much) smaller key sizes than for instance RSA. I did not find an open paper at this time**

The following section is an attempt of a summary but it is outside my area of expertise and understanding, maybe @joigus or other experts can contribute:
While Wiles' proof itself is not directly related to cryptographic applications, the deeper understanding and advanced techniques developed through his work have influenced fields that use elliptic curves. 
Without Wiles' Proof the theoretical framework supporting ECC would be less robust. The lack of proof of the Modularity Theorem* could leave gaps in understanding the deep properties of elliptic curves, potentially undermining confidence in their security properties and/or making them more vulnerable to sophisticated mathematical attacks.

*) https://en.wikipedia.org/wiki/Modularity_theorem

**) This seems interesting but I could only get the abstract: https://link.springer.com/chapter/10.1007/978-981-99-3758-5_5 

Quote

The research compares the key size and security level of the ECC and RSA algorithms and evaluates their suitability for usage in resource-constrained fog computing environments. ECC, a newer technique, offers the same level of security as RSA but uses smaller key sizes, making it more resource-efficient. 

 

[joigus]

On 7/5/2024 at 12:19 AM, Ghideon said:

That is, as far as I know, correct. There are some indirect applications in for instance ECC (Elliptic Curve Cryptography) and Elliptic Curve Digital Signature Algorithm (ECDSA). This may be of interest to OP since it is used, maybe on a daily basis, when browsing the internet; Elliptic Curve Cryptography (ECC) is of interest because it offers the same level of security with (much) smaller key sizes than for instance RSA. I did not find an open paper at this time**

Very interesting, thank you!

I'm very far from being an "expert" on cryptography, btw.