Does "Fermat's Last Theorem" hold for the Gaussian integers?
I don't know the answer to this question, but there are many
solutions in the form A + B sqrt(d) for various rational (non-
square) values of d. For example, with d=-3/2 we have
(1 + 5sqrt(d))^3 + (5 + 3sqrt(d))^3 = (3 + 4sqrt(d))^3
On the other hand, we know there are no solutions for d = 1.
To help decide whether or not solutions exist for a particular
exponent p (but not for the general case), it turns out that
Kummer's methods of 1850 can be applied to arbitrary rings
of algebraic integers (cf. Encycl Dict of Math, 2nd ed.) In
general, if p is an odd prime and u is a primitive pth root of
unity, let h denote the class number of the cyclotomic field
Q(u). In these terms, we can say that Case I of FLT in this
ring corresponds to the impossibility of
x^p + y^p + z^p = 0 , gcd(xyz,p)=1
whereas Case II corresponds to the impossibility of
x^p + y^p = k (1-u)^(np) z^p , gcd(xyz,p)=1
for some non-negative rational integer n and k is a unit of
Q(u). Kummer proved that if p does not divide the class number
h, then neither of these two equations has a solution. (These
are analogous to the "regular primes" in Kummer's theorem on FLT
for rational integers, but of course we need to determine the
set of regular primes for the particular algebraic ring in
question.) I don't know if Wiles' general proof of FLT for
rational integers covers any of these other algebraic rings.
Of course, if we allow the exponent p to divide xyz, then for some
values of d we know immediately that solutions exist. For example,
there are certainly solutions in Q(sqrt(5)) for the exponent n=3,
such as
(5 - 9sqrt(5))^3 + (12sqrt(5))^3 = (5 + 9sqrt(5))^3
In general we'll always have a solution of the form
(A - B sqrt(d))^3 + (C sqrt(d))^3 = (A + B sqrt(d))^3
for any integer d expressible in the form
6BA^2
d = ------------
(C^3 - 2B^3)
This gives solutions for (at least) the following squarefree values
of d with absolute magnitude less than 100:
d A B C
---- ---- ---- ----
-89 89 36 42
-87 145 27 6
-86 14279 2752 1364
-59 1559 531 552
-51 1727 544 508
-47 5953 1504 20
-41 451 256 296
-31 31 243 306
-26 307 117 87
-23 -23 9 6
-15 11 5 2
-11 1705 972 666
-6 1 1 1
-5 5 4 2
-2 1 8 10
2 17 9 21
5 5 9 12
6 5 1 3
15 215 4 42
17 2159 36 390
33 11 4 6
43 473 4 50
58 8207 8 382
69 1265 27 156
82 41 1 5
85 85 1 8
93 31 1 4
These are also the values of d such that there is a solution of
the form
(A + B sqrt(d))^3 + (A - B sqrt(d))^3 = C^3
which correspond to the solutions of
2A(A^2 + 3dB^2) = C^3
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