Fermats Last Theorem |
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Pierre de Fermat was a French mathematician who, along with Rene Descartes, is regarded as one of the two leading mathematicians of the early 17th century. Born on August 17,1602 in Beaumont-de-Lomagne, France, Fermats methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the founder of differential calculus. You can read more about Fermats life and background at the Britannica website by clicking here. Fermat died in 1665 on January 12 at the age of 62. |
Fermat developed a number of significant mathematical theorems, but he was not anxious to publish their proofs. Arguably the most famous of his theorems is known as his last theorem to which he said he had a proof but it was too lengthy to show in the margin of a book in which he had stated the theorem. The theorem, commonly referred to as Fermats Last Theorem, remained unproven until the late 20th century when British mathematician Andrew Wiles published an accepted proof in 1994.
Fermats Last Theorem can be stated as in the following image:
Note that all terms in the theorem, x, y, z, and n, are integers; that is, they are whole numbers not having any fractional parts.
For the equation in question (xn + yn = zn), consider the following cases:
Case 1: n = 1
When n = 1, the equation xn + yn = zn becomes x + y = z which indicates the simple addition of two numbers. When two integers are added, the resulting sum is always also an integer, so that all terms in the equation, x, y, and z, are all integers. Thus, there are an infinite number of combinations of these terms that will satisfy the equation.
Case 2: n = 2
When n = 2, the equation xn + yn = zn becomes x2 + y2 = z2 which many will quickly recognize as the mathematical representation of the Pythagorean Theorem which is used to calculate the length of the hypotenuse of a right triangle. The Pythagorean Theorem states that the area of the square of the side that is the hypotenuse is equal to the sum of the areas of the squares of the other two sides.
The simplest form of a right triangle where all sides of the triangle are integers is the 3-4-5 right triangle, where the two adjacent sides of the right angle have lengths of 3 and 4 and the hypotenuse has a length of 5; that is, x = 3, y = 4, and z = 5, so that x2 + y2 = 32 + 42 = 9 + 16 = 25 = 52 = z2. Since the lengths of the sides of the right triangle that are integer multiples of the 3-4-5 combination also satisfy the equation, there are again an infinite number of combinations of these terms that will satisfy the equation. For example, for x = 6, y = 8, and z = 10, we have that x2 + y2 = 62 + 82 = 36 + 64 = 100 = 102 = z2.
Case 3: n > 2; that is, n = 3, 4, 5, 6, ... etc.
This is the case where Fermats Last Theorem applies. And it says that there is no combination of integer values for x, y, and z that will satisfy the equation xn + yn = zn when n is an integer value greater than 2.
Using the modern technology of the digital computer, mathematicians have tried integer values for the components of the equation up to values of four million and have thus far not been able to find a combination that will satisfy the equation.
In other words, no one has thus far been able to disprove Fermats Last Theorem.
So, do you want to become instantly famous? Find a set of integer values of x, y, and z along with an integer value for n that is greater than 2 that satisfies the equation xn + yn = zn and you will have done what no one else has yet been able to do: you will have shown Fermat's Last Theorem to be false! Good Luck!!
Click the following links to read more about Fermat and his Last Theorem:
https://www.britannica.com/biography/Pierre-de-Fermat
https://en.wikipedia.org/wiki/Pierre_de_Fermat
https://www.vedantu.com/maths/fermats-last-theorem
https://mathshistory.st-andrews.ac.uk/Biographies/Fermat/
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