The equation in Fermat's Last Theorem, K^p + L^p = M^p can be transformed by dividing by M^p into (*) x^p + y^p = 1 where here x and y each lies between 0 and 1, and we'll let p be any positive real number. For fixed p, denote the locus of (*) in the unit square [0,1]x[0,1] by C_p. Then the curves {C_p | 0 < p < oo} continuously fill up the open square (0,1)x(0,1) and their endpoints lie on the boundary of the square. Questions: ---------- Let Q^2 denote the rational points in (0,1)x(0,1). 1) For which p > 0 is the intersection X_p = C_p \int Q^2 nonempty? Denote this set of p by N. (We know that p = 1 and p = 2 are included, and by FLT, no other positive integer.) 2) For which among p in N is X_p a dense subset of C_p ? Denote this subset of N by D. (We know that p = 1 and p = 2 are included.) 3) What other kinds of intersection sets X_p, if any, can occur for p in N-D ? —Dan Let p be a real number. The equation