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Pythagorean Triples

Pythagorean Triples. a 2 + b 2 = c 2. What are they?. Ex. (3, 4, 5)  3 2 + 4 2 = 5 2 (6, 8, 10)  6 2 + 8 2 = 10 2 Is that it? Whole Numbers? 1.5 2 + 2 2 = 2.5 2 a 2 + b 2 = c 2 with a, b, c in Q +. Primitive Pythagorean Triples. (3, 4, 5)  “Primitive”

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Pythagorean Triples

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  1. Pythagorean Triples a2 + b2 = c2

  2. What are they? Ex. (3, 4, 5)  32 + 42 = 52 (6, 8, 10)  62 + 82 = 102 Is that it? Whole Numbers? 1.52 + 22 = 2.52 a2 + b2 = c2 with a, b, c in Q+

  3. Primitive Pythagorean Triples (3, 4, 5)  “Primitive” a2 + b2 = c2 with a, b, c in Z+ and GCD(a, b, c) = 1 GCD(a, b) = 1 & GCD(a, c) = 1 & GCD(b, c) = 1 Proof: If k|a & k|b then a = kg, and b = kh with k, g, h, in Z+ a2 + b2 = (kg)2 + (kh)2 = k2g2 + k2h2 = k2(g2 +h2) = c2 Thus k|c, so (a, b, c) is NOT primitive

  4. Homework Week 2 (3, 4, 5)  (5, 12, 13)  (7, 24, 25),… (2n + 1, 2n2 + 2n, 2n2 + 2n + 1) with n in Z+ Can all three a, b, and c be even? Can only c be even?

  5. Plimpton 322 185412 + 127092 = 125002 How? (Babylon - 1900- 1600 BC)

  6. c2 – b2 c2 – b2 = a2 (c/a)2 – (b/a)2 = 1 α = c/a, and β = b/a  α2 – β2 = 1 = (α + β)(α – β) (α + β) = m/n and (α – β) = n/m where m > n in Z+ α = ½(m/n + n/m) = (m2 + n2)/2mn = c/a β = ½(m/n – n/m) = (m2 – n2)/2mn = b/a a2 + b2 = c2 a = 2mn, b = m2 – n2, and c = m2 + n2, with m and n relatively prime and m > n in Z+

  7. Homework Week 3 Terminating decimal (x, y) solutions for x2 + y2 = 1? Take line L  y = t(x + 1) x2 + (t(x + 1))2 = 1 x2 + t2(x2 + 2x + 1) = 1 x2 + t2x2 + 2xt2 + t2 = 1 (t2 + 1)x2 + 2t2x + (t2 – 1) = 0 x = -1 , and x = (1 – t2)/(1 + t2) y = 2t/(1 + t2) ([(1 – t2)/(1 + t2)], [2t/(1 + t2)])

  8. More fun! Take ([(1 – t2)/(1 + t2)], [2t/(1 + t2)]) Let t = n/m with n, m in Z+ x = (m2 – n2)/(m2 + n2) and y = 2nm/(m2 + n2) [(m2 – n2)/(m2 + n2)]2 + [2nm/(m2 + n2)]2 = 1 (m2 – n2)2 + (2nm)2 = (m2 + n2)2 a = m2 – n2, b = 2nm, and c = m2 + n2

  9. Cool Theorems If (a, b, c) is a primitive Pythagorean triple • then a and b have opposite parity • then c must be odd • then one of a or b is divisible by 3 • then one of a, b, or c is divisible by 5 • where b is even, then b is divisible by 4

  10. Integer Triangles If b is even, then the area is divisible by 6 (a, b, c) is primitive if and only if ∆ABC has the smallest area among the integer right triangles similar to ∆ABC. The length of the radius of the circle inscribed in an integer triangle is always an integer

  11. Interesting “Nature of Mathematics” by Karl Smith where a/b is reduced Ex. m = 3 82 + 152 = 172

  12. PSAT Use Ans 1 (2, 3) Ans 2 (18, 3)

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