20190510, 06:27  #1 
Romulan Interpreter
Jun 2011
Thailand
2×3×31×47 Posts 
FLT
No, this is not a tread for a new FLT proof
But if I found one, I will let you know first! I just want to ask a honest question: Before Sir Wiles proved FLT, what was the smallest prime for which we didn't know if FLT holds? Now, the answer is not easy. Reading wikipedia pages about FLT, modularity theorem, irregular primes, Wolstenholme primes, and other "monstrosities" like them, I suspect it was 16843^{(*)}. Due to the work of Sophie Germain, Ernst Kummer, Legendre, and others like them, the set of primes for which FLT was not known to hold was getting very thin. Before Legendre, the FLT was unknown for strong irregular primes like 67, 101, 149, but after he proved the first case of FLT for primes p such that at least one of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, and 16p+1 is prime, and all regular pairs were settled too, the smallest possible candidate was 263, which seems to be not only strong irregular, but all his "upper uncles and aunts" are composite (interestingly, its 6p+1 is prime). Was any theoretical result available that would clear the strong irregular primes between 263 and 16843? Note that with all due respect for Sam Wagstaff and co, I do not consider the "computational proofs" in this particular context, suppose I can not really understand (at this stage) how could you use the computer to prove that there is no solutions for, for example, a^3+b^3=c^3, unless you effectively find a solution and disprove the FLT.  ^{(*) }Now, related to 16843, you see this is like 16384, which is 2^{14} with some digits reversed... Is that a coincidence?^{(TM)} Do I get the "Enzocreti Award" for finding the similarity? Last fiddled with by LaurV on 20190510 at 13:17 Reason: s/16483/16843/g 
20190510, 07:43  #2 
Dec 2012
The Netherlands
1441_{10} Posts 
I don't have an exact answer to your question, but a good summary of how the story developed, including references to papers which may contain an answer,
is given in the 4th edition of the book "Algebraic Number Theory & Fermat's Last Theorem" by Ian Stewart and David Tall, published in 2016: https://www.crcpress.com/AlgebraicN.../9781498738392 
20190510, 08:16  #3  
Jun 2015
Vallejo, CA/.
31^{2} Posts 
Quote:
But 16483 = 53*311 16843 perhaps? Last fiddled with by rudy235 on 20190510 at 08:19 

20190510, 13:17  #4 
Romulan Interpreter
Jun 2011
Thailand
2·3·31·47 Posts 

20190510, 14:09  #5 
Jun 2015
Vallejo, CA/.
31^{2} Posts 

20190510, 17:34  #6  
Feb 2017
Nowhere
2^{3}·3·5·29 Posts 
Fermat’s Last Theorem said it had been raised to 4000000. The cited paper got in just under the wire.
Quote:
Last fiddled with by Dr Sardonicus on 20190510 at 18:16 Reason: rephrasing; corrections 

20190510, 18:38  #7  
Aug 2006
3×5^{2}×79 Posts 
Quote:


20190514, 06:34  #8  
Romulan Interpreter
Jun 2011
Thailand
2×3×31×47 Posts 
Quote:
Quote:
huh? 

20190514, 07:31  #9 
"Luke Richards"
Jan 2018
Birmingham, UK
288_{10} Posts 

20190514, 11:32  #10 
Feb 2017
Nowhere
2^{3}·3·5·29 Posts 
FLT had been proven for all odd prime exponents < 4000000; the next odd prime after 4000000 is 4000037.
The first case was proven for all primes < 8.858 x 10^{20}. The next prime after 885800000000000000000 is 885800000000000000009. Last fiddled with by Dr Sardonicus on 20190514 at 11:42 Reason: xifgin spoty 
20190514, 13:27  #11 
Aug 2006
13445_{8} Posts 
LaurV wants us to figure out what he means by "computational proofs" and exclude those; apparently this result is considered such.
