can you build or generate at least 5 pythagorean triples using one of many formulas? and can u verify each one of them in the pythagorean theorem equation
given any integers m > n,
a = m^2 - n^2
b = 2mn
c = m^2 + n^2
a,b,c form a Pythagorean triple
for (m,n) = (2,1) we get (3,4,5)
for (m,n) = (3,2) we get (5,12,13)
and so on
thanks
Certainly! Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
To generate Pythagorean triples, you can use one of the following formulas:
1. Euclid's Formula:
- For any positive integers m and n, where m > n:
- The three numbers of the triple are a = m^2 - n^2, b = 2mn, and c = m^2 + n^2.
- For example, if we choose m = 2 and n = 1:
- a = (2^2) - (1^2) = 3, b = 2(2)(1) = 4, c = (2^2) + (1^2) = 5.
- To verify, we can check if a^2 + b^2 = c^2: 3^2 + 4^2 = 9 + 16 = 25 = 5^2, which is true.
2. Pythagorean Triple Generator Formula:
- For any positive integer k:
- The three numbers of the triple are a = k(m^2 - n^2), b = 2kmn, and c = k(m^2 + n^2), where m > n > 0.
- For example, if we choose k = 2, m = 3, and n = 1:
- a = 2(3^2 - 1^2) = 16, b = 2(2)(3)(1) = 12, c = 2(3^2 + 1^2) = 20.
- To verify, we can check if a^2 + b^2 = c^2: 16^2 + 12^2 = 256 + 144 = 400 = 20^2, which is true.
Using these formulas, let's generate and verify five Pythagorean triples:
1. Using Euclid's Formula with m = 2 and n = 1:
- a = 3, b = 4, c = 5.
- Verification: 3^2 + 4^2 = 9 + 16 = 25 = 5^2, true.
2. Using Euclid's Formula with m = 3 and n = 2:
- a = 5, b = 12, c = 13.
- Verification: 5^2 + 12^2 = 25 + 144 = 169 = 13^2, true.
3. Using Euclid's Formula with m = 4 and n = 3:
- a = 7, b = 24, c = 25.
- Verification: 7^2 + 24^2 = 49 + 576 = 625 = 25^2, true.
4. Using the Pythagorean Triple Generator Formula with k = 2, m = 3, and n = 1:
- a = 16, b = 12, c = 20.
- Verification: 16^2 + 12^2 = 256 + 144 = 400 = 20^2, true.
5. Using the Pythagorean Triple Generator Formula with k = 3, m = 5, and n = 2:
- a = 63, b = 60, c = 87.
- Verification: 63^2 + 60^2 = 3969 + 3600 = 7569 = 87^2, true.
These are five examples of Pythagorean triples generated using the formulas, and their verification using the Pythagorean theorem shows that they are indeed valid.