Pythagorean theorem

The Pythagorean theorem is possibly the most well known mathematical theorem. It states that squaring the lengths of the two smaller sides of a right triangle and adding them together will get the length of the hypotenuse squared. This is expressed mathematically as:

<math>a^2+b^2=c^2</math> or, <math>c=\sqrt{a^2+b^2}</math>

This identity only holds true for a right triangle drawn on a flat (Euclidean) plane

The Pythagorean Theorem was developed by the Greek mathematician Pythagoras. The theorem describes a mathematical relationship between the lengths of the sides of a right triangle which can be illustrated as follows:

<math>\to a^2+b^2=c^2</math>

Many different proofs of the Pythagorean theorem have been devised. Euclid's proof is one of the most complicated and least intuitive.

One proof, below, appeared in an ancient manuscript with no explanation other than the word "look!" Essentially, the triangles in the large square, the hypotenuses of which form the border of the small square, form the two congruent rectangles in the upper right hand and lower left hand corners of the diagram. Therefore, the other two sections of the large square in this portion of the diagram possess the same area as the smaller square in the first diagram. The area of one the two small squares in the second portion of the diagram is given by <math>a^2</math>, and the other by <math>b^2</math>, therefore, the area of the small square in the first portion of the diagram is given by <math>a^2+b^2=c^2</math>, the Pythagorean theorem.

Corollary: No set of three lines with prime numbers as lengths can form a right triangle

Indirect proof: Let's describe a situation where lines with the lengths of three prime numbers do form a right triangle and see what follows.

Conceptualization

Let's call the two prime legs l₁ and l₂ and h the prime hypoteneuse.

One definition of a prime number p is that it is a natural number with no positive integral factor f greater than one and less than itself.

Second hypothesis

Conditional sub-proof of indirect proof: Let's introduce a second hypothesis and see what follows from it.

Simple assumption: One or more sets of two natural numbers, a pair n and o, for example, exist that have a positive integral factor f apart from one and themselves and common to both of them.

Corollary to assumption: Through the definition of prime numbers such as described in the first "conceptualization" section, if any pair of positive integers n and o have an common integral factor f apart from one and themselves, they are not prime because they both have a non-prime type of positive integral factor; in fact it's the same one.

Conditional sub-proof to be proved: Assuming no pair of prime numbers matches the set having the conditions described, no set of three lines with prime numbers as lengths can form a right triangle.

That is, it is to be proved that the first proposition (the condition) leads to the second.