John Bibby wrote:
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Date: Sun, 25 Jul 1999 11:31:53 +0100
From: John Bibby <qed@enterprise.net>
Subject: [HM] Germain and Safe primes
5. Did Germain prove Fermat's Last Theorem for the case where n is
a Germain prime, or did she "just" show that if the Fermat equation holds
for a Germain prime, then x and y must be multiples of n? (I have seen
both stated in the literature.) Does anyone know where/whether this proof
was published?
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The answer lies inbetween!
Sophie Germain in fact had a grand plan for proving Fermat's Last Theorem
for all exponents, which she pursued a long way with considerable success
in the second decade of the nineteenth century. The scope and
significance of her work have remained unknown for almost two hundred
years. My colleague Reinhard Laubenbacher and I made a preliminary
presentation on this research in the History of Mathematics session at the
January, 1995 annual joint AMS-MAA meetings in San Francisco, and we are
preparing it for publication.
Her grand plan was based on attempting to produce, for each odd prime
exponent p of the Fermat equation, infinitely many auxiliary primes of the
form 2Np+1 satisfying certain conditions. The results for auxiliary
primes of this form mentioned in a recent post by Udai Venedem to
historia-matematica from Legendre's memoir are in fact all due to Germain
as well (in her manuscripts), even though the Legendre memoir does not
mention this. Legendre and Germain were probably collaborating closely on
all this work, and some of the results are probably due to both of them.
What are today called Germain primes (primes of the form 2p+1) are more
special, emerging just from a particular specialization (today called
Germain's Theorem) of a weak version of a stronger theorem which is
proved in her (voluminous) handwritten manuscripts. Her methods continued
to be used successfully to make great progress on what we call Case I of
FLT well up into the 1980s (sic). She herself proved something a little
stronger than what we today call Case I up through exponent 100.
Many of her results were rediscovered in the late nineteenth century by
people unaware of her work, which was never published. The only
"publication" was Legendre's mention in a footnote in the second
supplement to the second edition of his Theorie des Nombres (also
published as a French Academy Memoir), to the effect that a result he
presents on the way to proving FLT for exponent 5 is due to Mlle. Sophie
Germain; this is essentially the theorem mentioned above. In 1819 she
wrote a letter to Gauss (unpublished) in which she outlined her entire
plan, and explained what progress she had made, which included detailed
analysis and calculations for applying her theorems and methods to many
exponents, in particular through exponent 100. For instance, she showed
that any solutions (not just those not divisible by 5!) to the Fermat
equation for exponent 5 would have to be at least 30 decimal digits in
size (and she didn't even use her TI-92!). Not only was her "theorem"
(really several theorems in her manuscripts) the first general result
about FLT, but her attack was much more comprehensive and far-reaching
than has heretofore been known.
It also looks promising that her manuscripts may show that, for at least
one exponent, she was actually the first to prove the full Fermat's Last
Theorem itself, but we are not claiming this yet. Many of her manuscripts
are difficult to read and decipher, since they were never prepared for
publication. Some even look like the scribblings of ourselves or our
students.
One missing link is the correspondence between Legendre and Germain, which
we know, from the only snippet we have, would shed invaluable additional
light, since Legendre was working on many of the same ideas. In fact, it
is now tantalizingly unclear just how much of the detailed analysis of the
applications of Germain's "theorem" made by Legendre (after he credits
Germain for the "theorem" in the footnote) were in fact independently his
(as the placement of the footnote implies), rather than primarily or at
least equally hers. We fear their correspondence is largely lost, but we
would be FOREVER grateful if anyone knows of its existence!
Best wishes,
David
David Pengelley (davidp@nmsu.edu)
Mathematics, New Mexico State University, Las Cruces, NM 88003 USA
Tel: 505-646-3901=dept., 505-646-2723=my office; Fax: 505-646-1064