Pythagorean Triple

Pronunciation: /pɪˌθægəˈriən ˈtrɪpəl/ ?

A Pythagorean triple is a set of three positive integers which satisfy the Pythagorean equation (A2 + B2 = C2). To 'satisfy' an equation means that the values can be substituted into the equation and the equation remains true. The smallest of the Pythagorean triples is (3,4,5). To verify that this triple is a Pythagorean triple, substitute the numbers into the Pythagorean equation: 32 + 42 = 52. Now simplify the exponents: 9 + 16 = 25. Finally, simplify the addition: 25 = 25. Since the equation 25 = 25 is true, (3,4,5) qualifies as a Pythagorean triple.

A primitive Pythagorean triple is a triple where a, b, and c are coprime. Integers are coprime if they have no common factors greater than 1.

History of Pythagorean Triples

The Pythagorean Theorem was one of the earliest achievements in analytic geometry. The ancients, however, had not discovered real numbers. They had only integers and rational numbers to work with. Ancient mathematicians explored Pythagorean triples as a way of finding integer values for the lengths of the sides of right triangles.

Generating Pythagorean Triples

Euclid’s Formula for Generating Pythagorean Triples

In exploring the possibilities of Pythagorean triples, a number of formulas were invented that allowed mathematicians to quickly find many Pythagorean triples. One of these formulas is called Euclid’s formula for generating Pythagorean triples. It is: A = 2mn, B = m2- n2, and C = m2+ n2, where m and n are any positive integers such that m > n, A and B are the lengths of the legs of the right triangle, and C is the length of the hypotenuse of the right triangle.

To show that these formulas work, substitute the formulas into the Pythagorean equation:

Equation(s)Description
A2 + B2 = C2Start with the Pythagorean equation.
2mn2 + (m2-n2)2 = (m2+n2)2Substitute 2mn in for A, m2 - n2 in for B, and m2 + n2 in for C.
4m2n2 + m4 - 2m2n2 +n4 = m4 + 2m2n2 + n4Expand the exponents.
m4 + 2m2n2 + n4 = m4 + 2m2n2 + n4Combine like terms. Since both sides of the equations are the same, this is an identity. This means that the formula will always generate Pythagorean triples.
Table 1: Euclid's formula for generating Pythagorean triples

While this formula will not find all Pythagorean triples. It will find infinitely many. The following table shows some of the Pythagorean triples that can be found using this formula:

mnABCVerification
212·2·1 = 422 - 12 = 4 - 1 = 322+ 12 = 4 + 1 = 542 + 32 = 52
16 + 9 = 25 →
25 = 25
312·3·1 = 632 - 12 = 9 - 1 = 832+ 12 = 9 + 1 = 1062 + 82 = 102
36 + 64 = 100 →
100 = 100
322·3·2 = 1232- 22 = 9 - 4 = 532+ 22 = 9 + 4 = 13122 + 52 = 132
144 + 25 = 169 →
169 = 169
412·4·1 = 842- 12 = 16-1 = 1542+ 12 = 16 + 1 = 1782 + 152 = 172
64 + 225 = 2289 →
289 = 289
Table 2: Examples of Euclid's formula for generating Pythagorean triples

Multiples of Pythagorean Triples

When one has found a Pythagorean triple, say (a, b, c), other Pythagorean triples can be found in the form (ka, kb, kc) where k is an integer greater than 1. However, Pythagorean triples generated by this method are not primitive. This can be proved by putting ka, kb, and kc into the Pythagorean theorem:

StepEquationsDescription
1(ka)2 + (kb)2 = (kc)2 ?This is the claim.
2k2a2 + k2b2 = k2c2Apply the distributive property of exponents.
3k2(a2 + b2) = k2c2Apply the distributive property of multiplication over addition and subtraction.
4(k2/k2)(a2+b2) = (k2/k2)c2Use the multiplicative property of equality to multiply both sides by 1/(k2).
51(a2+b2) = 1(c2)Use the identity (m/m) = 1, m ≠ 0 to replace (k2/k2) with 1.
6a2+b2 = c2Apply the property of multiplying by 1. The equation is now identical to the Pythagorean theorem.
Table 3: Proof of that multiples of Pythagorean triples are also Pythagorean triples.

For example, take the Pythagorean triple (3,4,5). Some Pythagorean triples that can be derived from this triple by multiplication are (6,8,10), (9,12,15), (12,16,20), …. A Pythagorean triple that is not a multiple of another Pythagorean triple is called a primitive Pythagorean triple. The first sixteen primitive Pythagorean triples are:

( 3 , 4 , 5)( 5, 12, 13)( 7, 24, 25)( 8, 15, 17)
( 9, 40, 41)(11, 60, 61)(12, 35, 37)(13, 84, 85)
(16, 63, 65)(20, 21, 29)(28, 45, 53)(33, 56, 65)
(36, 77, 85)(39, 80, 89)(48, 55, 73)(65, 72, 97)

Generating All Pythagorean Triples

In 1934 B. Berggren shows that all primitive Pythagorean triples can be generated from the triple (3, 4, 5). This recursive algorithm generates three Pythagorean triples for each triple it starts with. Each of those three triples then generates three more. If a and b are the legs of a Pythagorean triple and c is the hypotenuse:

new side anew side bnew side c
T1:a - 2b + 2c2a - b + 2c2a - 2b + 3c
T2:a + 2b + 2c2a + b + 2c2a + 2b + 3c
T3:-a + 2b + 2c-2a + b + 2c-2a + 2b + 3c
Table 4: Berggren's algorithm for generating Pythagorean triples.

Example

Start with (3, 4, 5).

new side anew side bnew side c
T1:3 - (2×4) + (2×5) = 5(2×3) - 4 + (2×5) = 12(2×3) - (2×4) + (3×5) = 13
T2:3 + (2×4) + (2×5) = 21(2×3) + 4 + (2×5) = 20(2×3) + (2×4) + (3×5) = 29
T3:-3 + (2×4) + (2×5) = 15-(2×3) + 4 + (2×5) = 8-(2×3) + (2×4) + (3×5) = 17
Table 5: Example of Berggren's algorithm for generating Pythagorean triples.

Properties of Pythagorean Triples

The properties of a primitive Pythagorean triple include:

  • (c - a)(c - b)/2 is always a perfect square. This is particularly useful in checking if a given triple of numbers is a Pythagorean triple.
  • Exactly one of a, b is odd; c is odd.
  • The area (A = ab/2) is an integer.
  • Exactly one of a, b is divisible by 3.
  • Exactly one of a, b is divisible by 4.
  • Exactly one of a, b, c is divisible by 5.
  • At most one of a, b, c is a square.
  • There exist infinitely many primitive Pythagorean triples.

References

  1. Gareth A. Jones, Josephine M. Jones. Elementary Number Theory, Corrected edition, pp 219-220. Springer, January 14, 1998.

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Pythagorean Triple. 2010-03-20. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/pythagoreantriple.html.

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2010-03-22: Initial version (McAdams, David.)

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