Constructing the Integer Solutions of . Senior Seminar Project By Santiago Salazar. Pythagorean Theorem. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. The Pythagorean Proposition by Elisha S. Loomis (1852 - 1940).
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Senior Seminar Project
By Santiago Salazar
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
A Pythagorean triangle is a right-angled triangle whose side’s lengths are positive integers.
As a result of the Pythagorean Theorem and the previous definition, our problem of constructing the integer solutions to the equation is equivalent to that of finding all Pythagorean triangles, whenever we restrict ourselves to positive integer solutions only.
We say that divides (denoted by ) if and only if there exists an integer such that .
We say that is greatest common divisor of and (denoted by ) if and only if
(i) and , and
(ii) if and , then .
We say that and are relatively prime if and only if .
If , then .
If and , then and .
We say that a positive integer solution to the equation i.e. a Pythagorean triple or equivalently an element of ) in which is a fundamental solution.
The recovery process is as follows:
If is a fundamental solution, then exactly one of and is even and the other is odd.
If is a fundamental solution, then is odd.
If and , then both and are squares.
Suppose that is a fundamental solution and is even. Then there are positive integers and with , , and such that
then . If in addition, , and are positive, , and , then is a fundamental solution.
where is even is a fundamental solution if and only if there are positive integers and with , , and such that
If , then
Recall that a Pythagorean triple generates a Pythagorean triangle with legs and and hypotenuse (according to our definition). Now given a Pythagorean triple , we define its area and perimeter by
Notice that these agree with the geometrical areas and perimeters of the corresponding Pythagorean triangle.