provided other modern mathematics. It is worth mentioning that the Taniyama- Shimura conjecture is a vital point in the development of the proof of Fermat’s last theorem. Taniyama-Shimura conjecture states that every elliptic curve over the rational numbers is uniformed by a modular form. So to prove Fermat’s last theorem, wiles had to prove the Taniyama-Shimura conjecture. Last but not least, the biggest similarity between the proof for Fermat’s last theorem and Shackleton’s is duration. Wiles spent 7 years to prove the theorem and Shackleton spent 18 years to explore the South Pole. It is really hard to imagine that such a simple theorem can cost over 400 hundred years to generate a book long solution. Here I will demonstrate the proof for n equal to 3.
Proof
Note: the mathematical argument below follows closely to the source 9 . However, the explanations are all in my own words.
First of all, we assume that the three non-zero integers x , y and z which are
pairwise coprime （ greatest common divisor is 1 ） and not all positive are the
3 + 3 + 3 = 0
solution to the equation
Let us assume that x , y and z are all odds and 3 , 3 and 3 are clearly all odds, hence the sum of them is odd. But this is obviously not true as their sum should be zero. First of all, x , y and z cannot be all even because they are coprime. Therefore, there is at least one even and one odd number. Furthermore, if the third number is even, 3 + 3 + 3 is not zero again. Hence we come to the conclusion that there are two odd numbers and one even number in x , y and z.
Hence Z may be assumed to be even to prevent loss of generality.
If = , substitute this into the original equation, we get 2 3 = 3 , which implies that x is even, so this is a contradiction.
Since x doesn’t equal y and they are both odd, their sum and difference are all even.
We express + 2 = 2 , − = 2 here u and v are coprime and one of them is even and the other is odd.
9 Proof of Fermat’s Last Theorem for specific exponents (2015) in Wikipedia. Available at: http://en.wikipedia.org/wiki/Proof_of_Fermat’s_Last_Theorem_for_specific_exponents (Accessed: 1 May 2015)
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