ON THE FIRST CASE OF FERMAT'S LAST THEOREM ... - Project Euclid
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ON THE FIRST CASE OF FERMAT'S LAST THEOREM ... - Project Euclid
w = 5), Frobenius [4], Pollaczek [5], Morishima [ó], and Rosser [7] for all prime values of m ^41. Wieferich's criterion alone has been applied by Meissner [8] and.
ON THE FIRST CASE OF FERMAT'S LAST THEOREM D. H. AND EMMA LEHMER
In 1909 Wieferich [l] proved his celebrated criterion for the first case of Fermat's last theorem, namely: The equation xp + yp = zp,
(1)
x, y, z prime to p,
has no solutions unless (2)
2 H s 1 (mod p2).
Since that time numerous other criteria of the form (3)
mp~x s 1 (mod p2)
have been proved by Mirimanoff [2] (for m = 3), Vandiver [3] (for w = 5), Frobenius [4], Pollaczek [5], Morishima [ó], and Rosser [7] for all prime values of m ^ 4 1 . Wieferich's criterion alone has been applied by Meissner [8] and Beeger [9] for p< 16,000 and was found to be satisfied only for p— 1,093 and 3,511, both of which cases failed to satisfy Mirimanoff's criterion. Until recently no effort has been made to combine these various criteria in a practical way. Mirimanoff observed, however, in 1910 that his criterion and that of Wieferich could be combined to state that equation (1) has no solutions for all primes p of the form 2a3/3± 1 or | 2 « ± 3 " | . In the presence of more criteria this statement can be extended thus : We call a number an "An number" (after Western) if it is divisible by no prime exceeding the nth prime pn. If the criterion (3) has been established for all m^pny then equation (1) does not hold if p is the sum or difference of two An numbers [l0]. Since all the numbers less than pn+i are An numbers, we may state that equation (1) has no solution for any prime in a region where the A n numbers are so dense that they do not differ by more than 2pn+i — l. This method was used in 1938 by A. E. Western [ l l ] to show that (1) is impossible for 16,000 Hence if n(x) denotes the number of A n numbers not exceeding x, then UP2/2)
(5)
£(p-
l)/2,
a condition which cannot hold for small p's if the An numbers are sufficiently dense. Other inequalities of this kind can be derived by separating the solutions of (4) into classes. We shall do this here only in the case in which the solutions are distinguished by their parity. For every positive odd solution o)/3) + PÎ2(log p*/3) ^{p-
l)/2.
This holds only for p> 102,108,200. Hence the first case of Fermat's last theorem is now proved for £
