Question

Examine this set of Pythagorean triples from part C. Look for a pattern that is true for each triple regarding the difference between the three values that make up the triple.

Describe this pattern. Then see if you can think of another Pythagorean triple that doesn’t follow the pattern you just described and that can’t be generated using the identity (x2 − 1)2 + (2x)2 = (x2 + 1)2. Explain your findings.

x-value Pythagorean Triple
3 (6,8,10)
4 (8,15,17)
5 (10,24,26)
6 (12,35,37)

Answer 1

• The two larger numbers differ by 2
• The smaller is the root of twice the sum of the larger two

Explanation:

The given formula gives rise to the above observations. it tells you ...

• a = 2x
• b = x^2 -1
• c = x^2 +1

This is a special case of Euclid's formula ...

• a = 2mn
• b = m^2 -n^2
• c = m^2 +n^2

for n=1. Since m and n can be any integer values, there are certainly many other Pythagorean triples that do not match the formula given in the problem. Using n=2, for example, the triples are ...

• a = 4x
• b = x^2 -4
• c = x^2 +4

Some triples from this set of formulas are (5, 12, 13), (20, 21, 29), (28, 45, 53), (36, 77, 85). Triples from this set of formulas will not match directly those from the formulas in the problem statement, but triples from both lists may reduce to the same primitive triple.