Wikipedia:Reference desk/Archives/Mathematics/2023 September 24

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September 24

at least one identical vertex, one identical side...

Let W be the set of Convex Polygons. Let R be a subset of W consisting of those Convex Polygons with rotational Symmetry. Let F be the subset of W where f is in F when for every vertex v in F, there is another vertex u with the same measure as v *and* for every edge e in F, there is another edge d with the same measure as e. R is a subset of F, but is R equal to F?Naraht (talk) 01:07, 24 September 2023 (UTC)

I assume that the measure of a vertex is the interior angle at that vertex and the measure of an edge is the same as its length. The pentagon with vertices $(0,0),(2,0),(2,2),(1,3),(2,0)$ is in F but does not possess rotational symmetry.  --Lambiam 07:00, 24 September 2023 (UTC)

Lambian Yes, your assumption is correct and thank you for the counter example!Naraht (talk) 02:35, 25 September 2023 (UTC)

Smallest "prime" for which Fermat's last theorem was NOT proved

Hello. I'm French and I asked this question [1] in our Oracle. Now I ask the same question in your Reference Desk.
I've just reread the article on Fermat's Last Theorem. Before Andrew Wiles' proof, this theorem had been proved for almost all integers > 2. It had been proved for infinite families of primes... but not all. Hence several questions:
Q1) Do we know the smallest prime number for which this theorem was not proved?
Q2) Is the sequence of these very special primes (or part of it) available anywhere on the Internet?
Q3) Could I get some informations about this subject in the site OEIS? Thanks.Jojodesbatignoles (talk) 11:49, 24 September 2023 (UTC)

Fermat's last theorem is extremely difficult to prove for the primes p satisfying these three conditions simultaneously:

1. All of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, 16p+1 are composite (sequence A152625 in the OEIS)
2. p is Bernoulli irregular prime (sequence A000928 in the OEIS)
3. p is Euler irregular prime (sequence A120337 in the OEIS)

The first few such primes p are 263, 311, 379, 461, 463, 541, 751, 773, 887, 971, …

—— 223.141.74.3 (talk) 01:47, 25 September 2023 (UTC)

Translated with the help of www.DeepL.com/Translator (free version) Jojodesbatignoles (talk) 11:49, 24 September 2023 (UTC)

In July 1993 (so, before Wiles' final, correct proof was published in May 1995) this paper was published, noting that FLT was true for all primes below four million. So, at the very least, the answer to Q1 is greater than that. Double sharp (talk) 12:27, 24 September 2023 (UTC)

For the history of proving for FLT: (of course, not count p = 1 and p = 2)

1. p = 4
2. p = 3
3. p = 5
4. p = 14
5. p = 7
6. p is Sophie Germain prime (i.e. 2p+1 is prime)
7. p is Bernoulli regular prime
8. p = 59, 67, 74
9. p = 37 (thus complete all p <= 100)
10. p <= 256 (in fact only complete all p <= 216, but all primes between 216 and 256 are either Bernoulli regular prime or Sophie Germain prime)
11. p is Euler regular prime
12. p <= 618 (p = 619 is extremely difficult to prove, and 619 is both Bernoulli irregular and Euler irregular, but note that 619*4+1 is prime)
13. primes p such that at least one of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, 16p+1 is prime
14. p <= 125000 (125003 is Bernoulli irregular, but 2*125003+1 is prime, thus the smallest prime such that FLT was not proved at that time should not be 125003, but I don’t know what is the smallest prime p > 125000 which is both Bernoulli irregular and Euler irregular, and none of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, 16p+1 is prime)
15. primes p such that 2^(p-1) != 1 mod p^2
16. primes p such that 3^(p-1) != 1 mod p^2
17. primes p such that F_{p-(p|5)} != 1 mod p^2
18. primes p not dividing the numerator of the Bernoulli number B(p-3)
19. primes p such that q^(p-1) != 1 mod p^2 for at least one of q = 5, 7, 11, 13, 17, 19
20. primes p such that q^(p-1) != 1 mod p^2 for some prime q <= 89
21. primes p such that q^(p-1) != 1 mod p^2 for some prime q <= 113
22. all p

223.141.74.3 (talk) 01:59, 25 September 2023 (UTC)

It seems to be no primes not satisfying any of conditions 1 to 21, but currently there is still no proof. 223.141.74.3 (talk) 02:16, 25 September 2023 (UTC)

The primes with was “the smallest prime number for which this theorem not proved” in history are 3, 5, 7, 11 (when p=7 was proved), 13 (when Sophie Germain primes were proved), 37 (when Bernoulli regular primes were proved), 101 (when p<=100 were proved), 257 (when p<=216 were proved), 263 (when Euler regular primes were proved), 619 (when p<=618 were proved), 751 (when primes p such that at least one of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, 16p+1 is prime were proved), >125000 (when p<=125000 were proved), >10^14 (when non-Wieferich primes were proved), … 223.141.15.192 (talk) 04:49, 25 September 2023 (UTC)

It should be noted that some of these results only hold for the "first case" of two, as distinguished by Sophie Germain. Specifically, the first case is when all $x,y,z$ in the equation $x^{p}+y^{p}=z^{p}$ are relatively prime (in contrast to exactly one of $x,y,z$ being divisible by $p$.) Also, the history of proofs is complicated and murky.

Here's who proved each of the successive criteria for impossibility:

1. Fermat himself

2. Euler, 1760 (not published, and apparently there was some mistake in the proof, though apparently you can patch it up with techniques that Euler himself used?)

3. Dirichlet and Legendre, 1825

4. Dirichlet, 1832

5. Lamé, 1839

The following were all proved specifically for the first case:

6. Germain, 1823

7. Kummer, 1850

8. Possibly Kummer, 1858, although Vandiver, 1920a and 1920b apparently said that Kummer made an invalidating error, and also I'm not entirely sure what numbers Kummer proved.

9. Mirimanoff, 1909

10. ???

11. Vandiver, 1940

12. ???

13. Legendre, 1823 (in full; Germain independently proved, but never published, a similar statement with a few more conditions, proving the first case for many numbers.)

14. ???

15. Wieferich, 1909

16. Mirimanoff, 1910

17. Sun and Sun, 1992 (by definition, non-Wall–Sun–Sun primes)

18. Genocchi, 1852 (note that this implies case I precisely for non-Wolstenholme primes)

19. Vandiver, 1914 for 5; Frobenius, 1914 for 11, 17 (possibly, apparently there were errors); possibly Rosser for the remaining based on MathWorld.

20. Granville and Monagan, 1988

21. Suzuki, 1994

And of course, in full generality:

22. Wiles, 1995

GalacticShoe (talk) 23:29, 26 September 2023 (UTC)

18 was proved in 1852 but 8 and 9 was proved in 1858 and 1909? But none of the primes <= 100 are Wolstenholme primes. 220.132.230.56 (talk) 02:28, 27 September 2023 (UTC)

I'm not sure how it works either, there's probably some specific criterion that I'm missing, but as it stands that's what it seems to me that the listed articles say. GalacticShoe (talk) 03:16, 27 September 2023 (UTC)

• it seems like any insanely large Irregular primes, which have never been computationally checked directly to see if they could made exceptions to FLT, fit well into your question >> they are infinite in number, and themselves are known to have interesting connections to abstract algebra since the time of Ernst Kummer, for example- (https://oeis.org/A000928), prior to the Wiles' proof in 1995, the primes of the type Wall-Sun-Sun are also suspected to be the possible source of exceptions to FLT, although even after than 27 years later, these elusive primes still hide themselves very carefully and defy any searching efforts of the mathematicians. 2402:800:63BC:DB8D:DC20:578F:8D73:7C28 (talk) 15:48, 24 September 2023 (UTC)

  Not only Wall-Sun-Sun primes, but also Wieferich primes and Wolstenholme primes. 223.141.74.3 (talk) 01:40, 25 September 2023 (UTC)