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:
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:
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 , and the other by , therefore, the area of the small square in the first portion of the diagram is given by , the Pythagorean theorem.
Corollary: No 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.
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.
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 n1 and n2, 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 n1 and n2 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 proven: Assuming no pair of prime numbers can possess the qualities of the pair set with the described factor, no three lines having prime numbers as lengths can form a right triangle.
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.
The conceptualization of the Pythagorean theorem in the case of the three prime numbers, that is using the assumption for an indirect proof is therefore: l12 + l22 = h2.
1. Let's divide one of the two 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. So let's assign o1 as being equal to l12 and o2 as being equal to l22, to limit them to showing they are natural numbers like the n's in the "simple assumption":
- (o1 / h) + (o2 / h) = h.
The conceptualization of the simple assumption can be expressed as
- If (n1 / f) + (n2 / f) = n3 then n1 and n2 are not prime.