A Pythagorean Triple is a set of integers that satisfy the Pythagorean Theorem, c squared= a squared + b squared. 3,4,5 and 5,12,13 are just two of many examples. Find another, that is not a multiple of the one of the two given, using the following formula.

n squared- m squared,2mn, n squared+ m squared, where m,n E I and n>m

n^2-m^2, 2mn , n^2 + m^2 , where n > m will always produce a Pythagorean triple.

If n and m are also relatively prime, that is they have no common factor between them, then that triple will also be unique.

e.g. n= 12, m=7

12^2-7^2 = 144 - 49 = 95
2(12)(7) = 168
12^2 + 7^2 = 144 + 49 = 193

95, 168, and 193 are a PT , and are not a multiple of any other PT.

Just a question.. should I have 3 numbers and not only two?

To find another Pythagorean triple using the given formula, we can choose values for 'm' and 'n', where 'n' is greater than 'm', and plug them into the formula n² - m², 2mn, n² + m². Let's choose some values and compute the triple:

Let's take m = 2 and n = 3.
Substituting these values into the formula, we get:
n² - m² = 3² - 2² = 9 - 4 = 5
2mn = 2 * 3 * 2 = 12
n² + m² = 3² + 2² = 9 + 4 = 13

Therefore, the Pythagorean triple using the values m=2 and n=3 is 5, 12, 13. This is another valid triple that is not a multiple of 3, 4, 5 or 5, 12, 13.

To find another Pythagorean Triple using the formula n^2 - m^2, 2mn, n^2 + m^2, where n and m are positive integers and n > m, follow these steps:

1. Choose a value for n. Let's start with a small value, such as n = 2.
2. Choose a value for m that is less than n. Let's try m = 1.
3. Plug the values of n and m into the formula to calculate the triple:
a = n^2 - m^2 = 2^2 - 1^2 = 3
b = 2mn = 2 * 2 * 1 = 4
c = n^2 + m^2 = 2^2 + 1^2 = 5
So the triple is 3, 4, 5.
4. Check if this triple is a multiple of the given triples (3, 4, 5) and (5, 12, 13). In this case, it is not a multiple, so we can continue.
5. To find more triples, try different values of m, keeping n constant. For example, let's try m = 2.
a = n^2 - m^2 = 2^2 - 2^2 = 0
b = 2mn = 2 * 2 * 2 = 8
c = n^2 + m^2 = 2^2 + 2^2 = 8
So the triple is 0, 8, 8.
6. Again, check if this new triple is a multiple of the given triples (3, 4, 5) and (5, 12, 13). Since it is a multiple of (3, 4, 5), we discard this triple.
7. Continue trying different values of m until you find a triple that is not a multiple of the given ones.

By following these steps, you can find another Pythagorean triple that is not a multiple of the ones given, using the formula n^2 - m^2, 2mn, n^2 + m^2.