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2HXY algebraic integer algebraic numbers applied arithmetical Bachet Bachet's equation y2 binary cubic class number coefficients complex numbers consider y2 cubic curve cubic equation Cxy2 deduced Diophantine Equation y2 Diophantos divisor equa equation f(x equation is impossible equation xn Euler Fermat wrote Fermat's last theorem Fermat's statement field R(d finite basis finite number form 4n found some results Frenicle Fueter fundamental solution Gauss genus unity Hence infinite descent infinity of rational integer solutions integer values invariant are equivalent mathematicians mathematics method of infinite mod 9 Nagell number of equations number of forms number of integer number of values Pellian equation T2 Pepin perfect cube positive integers prime to 2kf prove Fermat's last R(Jm rational numbers rational point P1 rational solutions set of numbers Smith's Prize solu solution of y2 solutions of Bachet's solve square free THEORY OF NUMBERS theory of quadratic tion unique fac unique factorization