The Extension of Fermat’s Last Theorem Zhiqiang Zhang School of Mathematics and Computer Science, Shanxi Datong University, Datong, 037009, China E-mail:
[email protected]
Abstract A new mathematical discovery is presented, we extend Fermat’s last theorem to make it to be a more general situation. In our discovery, the Fermat’s last theorem is only the special case. Keywords: Fermat’s last theorem, conjecture, extension
1
Introduction
As we known, Fermat’s Last Theorem or Fermat’s conjecture states that no three positive integers x, y and z can satisfy the equation z n = xn + y n
(1)
for any integer value n > 2. The above result was first discovered by Pierre de Fermat in 1637. This conjecture stimulated the development of algebraic number theory and it was known as one of the most difficult mathematical problems. It was successfully proved until Andrew Wiles released his work in 1994, which was published in 1995.
1
2
Zhiqiang Zhang’s conjecture: the extension of Fermat’s Last Theorem
We find that y 3 = x31 + x32 + x33
(2)
has solutions of positive integers, and y 4 = x41 + x42 + x43 + x44 y 5 = x51 + x52 + x53 + x54 + +x55
(3)
······ have solutions of positive integers. Generally, we have the conjecture as follows. Zhiqiang Zhang’s conjecture The equation yk =
n X
xki
(4)
i=1
exists solutions of positive integers if k = n, (n = 2, 3, · · ·), that is there are n + 1 positive integers can satisfy the above equation when k = n. Furthermore, the equation (4) has no solutions of positive integers if k > 2n−1 , that is no n + 1 positive integers can satisfy the equation (4) if k > 2n−1 , (n = 2, 3, · · ·). For example, in the case of n = 3, that is for the equation y k = xk1 + xk2 + xk3 ,
(5)
no four positive integers can satisfy the equation (5) if k > 4. Remark As we can see Fermat’s Theorem is only the special case of above conjecture when n = 2. Fermat’s conjecture stimulated the development of algebraic number theory, we hope that our conjecture can make a contribution to the mathematical development.
References [1] Andrew J. Wiles. Modular Elliptic Curves and Fermat’ Last Theorem. Annals of Mathematics, 1995, 141: 443-551. [2] http://en.wikipedia.org/wiki/Fermat
2
