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Fermat's Last Theorem for “Almost All” Exponents

Fermat's Last Theorem for “Almost All” Exponents FERMAT'S LAST THEOREM FOR "ALMOST ALL" EXPONENTS D. R. HEATH-BROWN Fermat's Last Theorem—which we shall abbreviate to FLT—is the (as yet unproved) assertion that the Diophantine equation x" + y" — z" has no solutions in positive integers if 11 ^ 3. It would suffice to deal with the case in which n is prime, and this is where the most significant work has been done. None the less it is not yet known even whether FLT is true for infinitely many prime exponents. If one considers general exponents n one sees that FLT is true at least for a proportion 1-5 n 3 « p « 125000 of all n, since the cases n = 4 and n = p ^ 125000 (p =£ 2) are known to hold (Wagstaff [3]). The object of this note is to prove the following corollary to the recent work of Faltings [1] on Mordell's Conjecture. THEOREM. FLT is true for "almost all" exponents n. That is, ifN(x) is the number of n ^ x for which FLT fails, then N(x) = o{x) as x -*• 00. Unfortunately the proof, being based on Faltings' Theorem, is ineffective. Thus, given e > 0, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Fermat's Last Theorem for “Almost All” Exponents

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References (3)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/17.1.15
Publisher site
See Article on Publisher Site

Abstract

FERMAT'S LAST THEOREM FOR "ALMOST ALL" EXPONENTS D. R. HEATH-BROWN Fermat's Last Theorem—which we shall abbreviate to FLT—is the (as yet unproved) assertion that the Diophantine equation x" + y" — z" has no solutions in positive integers if 11 ^ 3. It would suffice to deal with the case in which n is prime, and this is where the most significant work has been done. None the less it is not yet known even whether FLT is true for infinitely many prime exponents. If one considers general exponents n one sees that FLT is true at least for a proportion 1-5 n 3 « p « 125000 of all n, since the cases n = 4 and n = p ^ 125000 (p =£ 2) are known to hold (Wagstaff [3]). The object of this note is to prove the following corollary to the recent work of Faltings [1] on Mordell's Conjecture. THEOREM. FLT is true for "almost all" exponents n. That is, ifN(x) is the number of n ^ x for which FLT fails, then N(x) = o{x) as x -*• 00. Unfortunately the proof, being based on Faltings' Theorem, is ineffective. Thus, given e > 0,

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1985

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