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Old 2019-05-10, 06:27   #1
LaurV
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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 214 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 2019-05-10 at 13:17 Reason: s/16483/16843/g
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Old 2019-05-10, 07:43   #2
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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/Algebraic-N.../9781498738392
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Old 2019-05-10, 08:16   #3
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Quote:
Originally Posted by LaurV View Post
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?
16483?

But 16483 = 53*311

16843 perhaps?

Last fiddled with by rudy235 on 2019-05-10 at 08:19
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Old 2019-05-10, 13:17   #4
LaurV
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Quote:
Originally Posted by rudy235 View Post
16483?
But 16483 = 53*311
16843 perhaps?
Yep, edited in all 3 places. Thanks.
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Old 2019-05-10, 14:09   #5
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Quote:
Originally Posted by LaurV View Post
Yep, edited in all 3 places. Thanks.
The best thing is that you do not lose your claim to the Enzocreti award!

Also, take into account that 16843 -16384 =459 and that 4+5=9
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Old 2019-05-10, 17:34   #6
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Fermat’s Last Theorem said it had been raised to 4000000. The cited paper got in just under the wire.
Quote:
There were many other refinements of similar criteria for Fermat’s Last theorem to be true. Computer calculations based on these criteria led to a verification that Fermat’s Last theorem is true for all odd prime exponents less than four million [BCEM], and that the first case is true for all l <= 8.858 x 1020 [Su].
<snip>
[BCEM] J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä,
Irregular primes and cyclotomic invariants to four million,
Math. Comp. 61 (1993), 151–153.
<snip>
[Su] J. Suzuki,
On the generalized Wieferich criterion,
Proc. Japan Acad. 70 (1994), 230-234.

Last fiddled with by Dr Sardonicus on 2019-05-10 at 18:16 Reason: rephrasing; corrections
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Old 2019-05-10, 18:38   #7
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Quote:
Originally Posted by Dr Sardonicus View Post
Fermat’s Last Theorem said it had been raised to 4000000. The cited paper got in just under the wire.
So the smallest unresolved exponent was 4000037, and for exponents under 885800000000000000009 it was known that the exponent must divide one of the terms.
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Old 2019-05-14, 06:34   #8
LaurV
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Quote:
Originally Posted by Dr Sardonicus View Post
Fermat’s Last Theorem said it had been raised to 4000000. The cited paper got in just under the wire.
Quote:
Originally Posted by CRGreathouse View Post
So the smallest unresolved exponent was 4000037, and for exponents under 885800000000000000009 it was known that the exponent must divide one of the terms.

huh?


Quote:
Originally Posted by LaurV View Post
Note that with all due respect for Sam Wagstaff and co, I do not consider the "computational proofs" in this particular context
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Old 2019-05-14, 07:31   #9
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Quote:
Originally Posted by LaurV View Post
Sir Wiles
English pedantic correction: Sir Andrew.
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Old 2019-05-14, 11:32   #10
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Quote:
Originally Posted by LaurV View Post
huh?
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 1020. The next prime after 885800000000000000000 is 885800000000000000009.

Last fiddled with by Dr Sardonicus on 2019-05-14 at 11:42 Reason: xifgin spoty
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Old 2019-05-14, 13:27   #11
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Quote:
Originally Posted by Dr Sardonicus View Post
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 1020. The next prime after 885800000000000000000 is 885800000000000000009.
LaurV wants us to figure out what he means by "computational proofs" and exclude those; apparently this result is considered such.
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