Fermat’s Last Theorem Pt.1
Discovery of Fermat’s Last Theorem
In 1637, a historic theorem was discovered by the French mathematician called Pierre de Fermat .
xn+yn=zn
For any integer value of n greater than 2, no three positive integers x, y, and z satisfy the equation.
He wrote in his copy of the Arithmetica that he had proven this theorem.
“Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere. Cuius rei demonstrationem mirabilem sane detexi banc marginis exiguitas non caperet.”
“It is impossible for a cube to be a sum of two cubes, a fourth power to be a sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”
This was a simple equation that has transformed Pythagorean Theorem of x2+y2=z2, by changing the exponent to any integer greater than 2. However, despite the small change given to the equation, it seemed impossible to find the solution for the equation xn+yn=zn(n3), as mentioned by Fermat in his annotation.
Therefore, this theorem named <Fermat’s Last Theorem> has remained unsolved for over 300 years.
Of course, there were countless attempts from various mathematicians to solve this problem, and accordingly there was some progress made throughout the years.
First Mathematician to Challenge the Theorem: Leonhard Euler
The first mathematician who challenged himself to prove this theorem and took a hopeful step towards solving this theorem was Leonhard Euler, a legendary mathematician from the 18th century. In order to prove that there is no solution of x, y and z for any integer value of n greater than 2, Euler used a method of finding the proof that there is no solution for any one of the equations then enlarging the scope of this logic.
x3+y3=z3
x4+y4=z4
x5+y5=z5
x6+y6=z6
x7+y7=z7
……
Fortunately, the start was favourable since another annotation was found in Fermat’s Arithmetica, which showed the proof that there is no solution for n=4. Although it also was an unfinished annotation with the same reason of “this margin is too small to contain it”, the logic of the proof of contradiction was distinctive, therefore Euler was able to reenact this proof.
To prove that there is no integer solution for n=4, Fermat first assumed that there are integer solutions for x4+y4=z4 and then tried to figure out the logical contradiction. For instance, since he had assumed that there are an integer solutions, it is supposed that:
x=X1 y=Y1 z=Z1
Because of the properties of these three integer solutions, the other integer solutions that have the smaller values (X2, Y2, Z2) should exist. And if the same logic is applied to those three integer solutions ⎯ X2, Y2, and Z2 ⎯ another smaller integer solutions of X3, Y3, and Z3 also must exist. This process of getting smaller integer solutions could be repeated infinitely, therefore it is concluded that “infinitely small” integer solutions can be obtained. However, due to the fact that x, y and z are integers, this endless repeating process of getting smaller integer solutions should be ended at some point ⎯ there are no infinitely small integers. Here, the contradiction is caused by the logic that this process can be repeated endlessly, thus proving that the hypothesis itself of equation x4+y4=z4 having integer solutions of x=X1 , y=Y1 and z=Z1 is wrong. Through this the one special case of Fermat’s Last Theorem ⎯ when n=4 ⎯ was verified.
With this verification as the starting point, Euler attempted to induce the proofs for every other equation (or every other n values). The first equation he tried was x3+y3=z3, which was the equation that had to be proven first for following the sequential order. In order to apply Fermat’s verification for n=4 to n=3 without making logical loopholes, Euler introduced the concept of imaginary number* (complex number specifically) and proved the theorem for n=3 using infinitely repeating proof of contradiction.
However, the method of Euler did not work out for n5, therefore Euler’s attempt to solve the theorem through applying the infinitely repeating proof of contradiction has failed. But still, Euler had left a partial but the first progress in solving Fermat’s Last Theorem.
* Imaginary number is what has the value of -1 when it is squared and has symbol of i
Throughout the contribution of Euler and Fermat himself, the circumstances got slightly better, because the proof for n=4 also proved equation for n=8,12,16,20, … (multiples of 4). The exponents with multiples of 4 can be expressed with a different relevant exponent that also has an exponent of 4. For example, X8 can be expressed as (X2)4. Therefore the logic that has applied for n=4 is equally applied for n=8,12,16,20, …. , leaving only equations with n value of prime numbers for proof, until the complete conquest of <Fermat’s Last Theorem>.
Second Mathematician to Challenge the Theorem: Sophie Germain
Then, 75 years after the achievement of Euler, the second person who had contributed to solving the theorem appeared — Sophie Germain, the female mathematician from France. She had provided a new way of solving by suggesting a special type of prime number to prove the n value generally, rather than proving n values one by one. The prime numbers she was interested in had a unique property of having another prime number as a result of multiplying the initial prime number itself with 2 and adding 1 — when p is a prime number, 2p+1 is also a prime number.
She expanded a logic that there is no solution for these special types of numbers of n, because this made at least one of the x, y or z values to be a multiple of n. This was a strong restriction given on the values of x, y and z because there was a very low possibility of the integer solution satisfying this to exist.
Later, in 1825, the proof that there is no solution for n=5 was made by two individual mathematicians from France — Peter Gustav Lejeune-Dirichlet and Adrien-Marie Legendre — using Germain’s logic. And furthermore, 14 years after that, Gabriel Lamé, another mathematician from France, proved for n=7 through adding more of his own method on Germain’s logic. So more and more equations for n with prime numbers had been proven by different mathematicians using Germain’s logic as the foundation.
Citations
SCIENCE PHOTO LIBRARY. “Pierre de Fermat, French Mathematician — Stock Image — H406/0229.” Science Photo Library, 2020, www.sciencephoto.com/media/225141/view/pierre-de-fermat-french-mathematician. Accessed 17 May 2022.
Grigg, Russell. “Euler — Creation.com.” Creation.com, 2017, creation.com/euler. Accessed 17 May 2022.
“Sophie Germain.” The MY HERO Project, 2022, myhero.com/S_Germain_dnhs_kt_US_2017_ul. Accessed 17 May 2022.
“Sophie Germain.” The MY HERO Project, 2022, myhero.com/S_Germain_dnhs_kt_US_2017_ul. Accessed 17 May 2022.
