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Pythagoras said, “Everything is number.” Gauss said, “Mathematics is the queen of science,and number theory is the crown of mathematics.” Hua Luo geng also said, “The universe is big,the particle is small, the rocket is fast, the chemical industry is clever, the earth is changing, the mystery of biology, the daily life is complex, and mathematics is used everywhere.”
The Fermatʼs Last Theorem is a famous mathematical theorem. In 1637 French amateur mathematician Fermat proposed and lost “I have a wonderful proof” and since has stumped the leading mathematicians of the world, In 1995 the British mathematician Wiles had proved it with modem profound mathematical methods.He cause a sensation of the world.Now that the Fermat Last Theorem（FLT） craze has passed, with the passage of time, many people find that even after more than 300 years, the original proof is still interesting to explore. Even Wiles had to resort to 20th-century techniques to prove a 17th-century conundrum. Did Fermat have “a really beautiful proof”? Some saw it as the flawed product of a genius in a rare, confusing moment; it is also argued that Fermat did have an original proof, probably using several arguments so cunning that everyone from Euler to Wiles missed it. Whatever the proof was, it must have been based on 17thcentury techniques, and my intuition was that Fermat had this extra ordinary talent and character.
My hobby is mathematics. In 1983, I became interested in Fermat’s conjecture. Believing that Fermat had a wonderful proof, I had searched for it for 30 years, working on it whenever I had a time. Unlike many modern mathematicians, I did not use the method of partial proof, but rather a common law for all the integers, and I took a different approach to Fermat's original mathematical methods to find his original answer. My idea was the same as that of the Norwegian mathematician Abel in the 17th century, but I went a few steps further until I had a result. Chapter 2 introduces the integer parameter U. I find that X, Y, and Z have independent and inter-related expressions with the help of U; in 2.1, the independent specific equations of X, Y and Z in the case of [I] are found,and then the interrelated general equation of [I] is obtained. In 2.2, the independent specific equations of X, Y and Z in case [Ⅱ] are found, and then the interrelated general equation[Ⅱ] is obtained. So we have two sets of equations in two cases, poetic in their symmetry. In Chapter 2.3,the root formula of Pythagorean equation is derived by using the above method, which proves that the above method is correct and feasible. But if you’re going to keep using all these identity transformations, you’re going to get nowhere, and you’re going to get stuck. In 2013, inspired by the theory of solving equations of higher order, the derivative of the above equation was introduced by Newton’s tangent method in 3.1 and 3.2 respectively, and the conditions were added.The answer was found from the recursive formula of root finding, and the result was deduced at last.
The purpose of this paper is to explore the “very beautiful proof” new method of proof. In the following paper, I will start from the new idea and solve the problem layer by layer with the mathematical methods of the seventeenth century. Proving this hypothesis is an amazing and interesting thing in itself . There should be an original proof , maybe it is not perfect, but with a little luck. However, today I still take out the manuscripts that have been stored in the drawer for many years to review and organize. The original intention of publishing this book is that I am willing to arouse the interest in mathematics together with the readers. I am not afraid to spend a lot of time and energy, or have some shortcomings in my thinking that are worth discussing.
提要 本文利用整数的性质對費马方程的二种情況进行推导，情況（Ⅰ）是p ∤ XYZ；情況（Ⅱ）是p│XYZ。推导得到二条相应的特殊形式的新的方程。然后利用牛顿切线法的递推公式，推导出收敛式的规律，或者与前设矛盾，或者导致无穷递降，所以费马方程永远无整数解得到了费马大定理的严格而又详细的证明。
关键词 恒等变换，牛顿切线法，递推公式，收敛公式，无穷递降法，最小数原理