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The Pythagorean question, x2+y2 = z2, has an infinite group of solutions (x, y, z), these solutions are geometric-related, which are regarded as Pythagorean groups, such as (3n, 4n, 5n), and (7n, 24n, 25n). From this point, the subject of this paper is based on the conception of Diophantine equation and one of its example—Fermat’s last theorem. Geometric and infinite descending methods will be considered mainly to address this problem. Although geometric method fails to deal with the equation, the bridge between triangle theory and algebra equation can be constructed, and infinite descending method is also useful to this problem. Additionally, some other conclusions will be found like other types of n, and some famous mathematicians’ ideas as an extended content. In the paper, the more simple and understandable methods are looked for even if it will be finally disproved. Next, the final results and conclusions about the Fermat’s last theorem point out that the case n=3, 4 can be solved by infinite descending method which creates smaller and smaller solutions in order to achieve a contradiction although that of geometric shifting can only deal with part of each case (n=3), due to the difficulty of coping with the case cos(a) is in (0, 1/2), so a further study should be operated or it is hard for people to deal with Fermat’s last theorem by triangle-related method by cosine rule. As for research conclusion, the infinite descending method is suitable for the cases of n = 3, 4; although, by creating new pairs of smaller solutions in various ways. Only the cases of cos(c) ∠ 0 when in geometric trials, Fermat’s last theorem can be solved, but other cases (cos(c) < 0) have difficulties in linking the characters of triangle with number theory, which failed.
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