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:

$a^{2}+b^{2}=c^{2}$ or, $c={\sqrt {a^{2}+b^{2}}}$

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:

$\to a^{2}+b^{2}=c^{2}$

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 $a^{2}$, and the other by $b^{2}$, therefore, the area of the small square in the first portion of the diagram is given by $a^{2}+b^{2}=c^{2}$, the Pythagorean theorem.

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

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

Conceptualization

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

One definition of odd prime number op is that it is an odd natural number with no positive integral factor f other than one and itself.

Hypothesis

Simple assumption: The hypoteneuse is always larger than either of its lengths.

That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the assumption for a conditional proof) leads to the second.

Conceptualization (cont'd)

The conceptualization of the Pythagorean theorem in the case of the three odd prime numbers, that is using the assumption for an indirect proof is therefore: l₁² + l₂² = h².

Reasoning

1. Let's divide one of the two odd prime number "h" factors contained in the "h²" by both sides of the equation:

(l₁² / h) + (l₂² / h) = h.

2. And since l₁ and l₂ are whole numbers, that which results when they are multiplied by themselves, namely l₁² and l₂², certainly are as well.

3. Because of the simple assumption, the square of the odd prime legs l₁² and l₂² only have a single odd prime factor h available with which to divide and therefore the leg lengths squared are not divisible by the odd prime hypoteneuse by the given definition of an odd prime number (otherwise it would give it a factor both greater than one and itself).

4. If the sum of square of the lengths happened to be a whole number divisible by some hypoteneuse, it must be divided by an odd prime hypoteneuse. Odd numbers take the form of 2n + 1, which when squared gives 4n² + 4n + 1, which also must be an odd number. Two odd numbers summed always gives an even number, which would result in the sum of square of the lengths not being fully divisible by any odd prime hypoteneuse.

5. Therefore the sum of each of the squared legs divided by an odd prime hypoteneuse can never be a odd prime number like the h that remains when h is divided through, contradicting the rephrased original premise of the indirect proof and demonstrating the opposite of the assumption to be true, namely the three lengths in a pythagorean triangle can never be all odd primes.