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The Pythagorean Triple Essay

Pythagoras a Greek philosopher and mathematician is very famous for its Pythagorean Theorem. This theorem states that if a, b and c are sides of a right triangle then a2 + b2 = c2 (Morris, 1997).

The study of the Pythagorean triples started long before Pythagoras knew how to solve it. There were evidences that Babylonians have lists of the triples written in a tablet. This would only mean that Babylonians may have known a method on how to produce such triples (Silverman, 16 – 26). Pythagorean triple is a set of number consisting of three natural numbers that can suit the Pythagorean equation a2 + b2 = c2. Some of the known triples are 3, 4, 5 and 5, 12, 13 (Bogomolny, 1996). How can we derive such triples?

If we multiple the Pythagorean formula by 2 then we generate another formula 2a2 + 2b2 = 2c2. This only means that if we multiply 2 to the Pythagorean triple 3, 4, 5 and 5, 12, 13 then we can get another set of Pythagorean triple. The answer to that is triple 6, 8, 10 and 10, 24, 26. To check whether the said triple are Pythagorean triple, we can substitute it to the original formula a2 + b2 = c2.

Check: is 6, 8, 10 Pythagorean triple?

62 + 82 = 102

36 + 64 = 100

100 = 100

Thus 6, 8 and 10 satisfy the Pythagorean equation.

Δ 6, 8, 10 is a Pythagorean triple.

Check: is 10, 24, 26 satisfy the Pythagorean equation?

102 + 242 = 262

100 + 576 = 676

676 = 676

Thus 10, 24, 26 satisfy the Pythagorean equation.

Δ 10, 24, 26 is a Pythagorean triple.

If we multiply the Pythagorean equation by 3 and using the first 2 Pythagorean triple mentioned above, we can yield another set of Pythagorean triple. Thus we can formulate a general formula that can produce different sets of Pythagorean triple. We can generate an infinite number of Pythagorean triple by using the Pythagorean triple 3, 4, 5. If we multiple d, where k is an integer, to that triple we will yield different sets of Pythagorean triple all the time.

d*(3, 4, 5) where d is an integer.

Check: if k is equal to 4 we get a triple 12, 16, and 20. Is this a Pythagorean triple?

By substitution,

122 + 162 = 202

144 + 256 = 400

400 = 400

Thus 12, 16, 20 satisfy the Pythagorean equation.

Δ 12, 16, 20 is a Pythagorean triple.

Check: if k is equal to 5 we get a triple 15, 20, 25. Is this a Pythagorean triple?

By substitution,

152 + 202 = 252

225 + 400 = 625

625 = 625

Thus 15, 20, 25 satisfy the Pythagorean equation.

Δ 15, 20, 25 is a Pythagorean triple.

But the formula given above is just a formula for getting the multiples of the Pythagorean triple. But is there a general formula in getting these triples? There are formulas that can solve each and every Pythagorean triple that one can ever imagine. One formula that can give us the triples is a = st, b = (s2 + t2)/2 and c = (s2 – t2)/2 (. A simple derivation of these formula will come from the main formula a2 + b2 = c2 (Silverman, 16 – 26). This is a shorten way to derive the formula from theorem 2.1(Pythagorean triples).

a2 + b2 = c2      with a is odd, b is even and a, b and c have no common factors.

a2 = c2 – b2      by additive property

a2 = (c – b)(c + b)       by factoring (difference of two squares)

by checking

32 = (5 – 4)(5 + 4) = 1*9

52 = (13 – 12)(13 + 12) = 1*25

72 = (25 – 24)(25 + 24) = 1*49


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