Generating Pythagorean triples using Euclid's formula

Pythagorean Triple is a term derived specifically from the theorem that a right triangle displays the equation. This theorem, named aptly the Pythagorean Theorem, states that "in any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides" (Mack, 2010). How do the two relate? A Pythagorean Triple is simply the term derived for a, b, and c, provided that 1) a, b, and c are positive integers and 2) a, b, and c fulfill the equation. Silverman's Number Theory chapter on "Pythagorean Triples" has highlighted the relevant proof for the equation of the Pythagorean Theorem. What further equations can be used to find subsequent Pythagorean Triples?

The primitive Pythagorean triple begins with set T (3,4,5). Already, set T. fulfills both the requirements 1) and 2) as stated above; that the numbers in the set are positive integers and, with the use of substitution (where a = 3, b = 4, and c = 5), they complete the Pythagorean equation. It becomes evident, then, that there is an infinite amount of Pythagorean triples, provided they are all integers and follow the formula. Euclid's formula states that one can generate Pythagorean triples by the following method:

Given positive integers p and q, such that p > q, there exists a, b, and c where a = p2 -- q2, b = 2pq, and c = p2 + q2 (Turner, 2006).

To test this formula out, one can use starting integers 1, and 2, where p = 1 and q = 2. Calculations: a = (2)2 -- (1)2 = 4 -- 1 = 3; b = 2(2)(1) = 4; c = (2)2 + (1)2 = 4 + 1 = 5

Thus, a = 3, b = 4, and c = 5. This triple is already known to fulfill the Pythagorean Theorem equation. One can continue to generate triples using the same formula with subsequent numbers.

Calculations

Example 1: p = 3, q = 2

Calculations: a = (3)2 -- (2)2 = 9 -- 4 = 5; b = 2(3)(2) = 12; c = (3)2 + (2)2 = 9 + 4 = 13

a = 5, b = 12, c = 13

Verification: 52 + 122 = 132 25 + 144 = 169-169 = 169 Check

Example 2: p = 3, q = 1

Calculations: a = (3)2 -- (1)2 = 9 -- 1 = 8; b = 2(3)(1) = 6; c = (3)2 + (1)2 = 9 + 1 = 10

a = 8, b = 6, c = 10

Verification: 82 + 62 = 102 64 + 36 = 100-100 = 100 Check

Example 3: p = 4, q = 2

Calculations: a = (4)2 -- (2)2 = 16 -- 4 = 12; b = 2(4)(2) = 16; c = (4)2 + (2)2 = 16 + 4 = 20

a = 12, b = 16, c = 20

Verification: 122 + 162 = 202-144 + 256 = 400-400 = 400 Check

Example 4: p = 4, q = 1

Calculations: a = (4)2 -- (1)2 = 16 -- 1 = 15; b = 2(4)(1) = 8; c = (4)2 + (1)2 = 16 + 1 = 17

a = 15, b = 8, c = 17

Verification: 152 + 82 = 172-225 + 64 = 289-289 = 289 Check

Example 5: p = 4, q = 3

Calculations: a = (4)2 -- (3)2 = 16 -- 9 = 7; b = 2(4)(3) = 24; c = (4)2 + (3)2 = 16 + 9 = 25

a = 7, b = 24, c = 25

Verification: 72 + 242 = 252 49 + 576 = 625-625 = 625 Check

Conclusion