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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