BLOG ENTRY #1

The question that am investigating is, Pythagorean triples are right triangles that have integer side lengths. how many of these are there? can you predict when they will occur?
When i began investigating i started by listing the the Pythagorean Triples that Ms.sheppherd-brick listed for us in the packet for week 27. I have not yet found out a pattern to figure out how you can predict when Pythagorean Triples will occur but did learn how you can find Pythagorean Triples with this equation,"take any 2 numbers, find 2 times their product, the sum of their squares, than the difference of their squares and the three different numbers that show up are going to be a Pythagorean Triple." (Diophantus) To prove that Diophantus's formula was correct I tried it out. Although the Pythagorean Triples 3-4-5 was giving to me in the week 27 packet, i also figured it out using Diophantus's formula.

1. 1*8*2= 4
   
2. 1^2+2^2= 1+4= 5
   
3. 2^2-1^2= 3
   

In the list below I figured out some Pythagorean Triples using his formula.

• 3-4-5
• 6-8-10
• 9-12-15
• 12-13-16
• 10-24-26
• 5-12-13
• 7-24-25
• 8-15-17
• 16-30-34
• 20-21-29
• 16-65-63
• 12-16-20
• 72-135-153
• 5-12-13
• 24-143-145
• 9-40-41
• 10-24-26

Although i didn't find a pattern when trying to point out Pythagorean Triples, i did notice that when checking to see if the 3 numbers were actually Pythagorean Triples i had to add the 2 smaller squared numbers and get the sum. Then i had to find the square of the largest number and in all my cases the sum and the squared was equal proving that they were Pythagorean Triples. Below is what i did to prove the Triples.(visually)

1. 4^2+3^2= 25
2. 5^2= 25

Since 3 and 4 were the smallest numbers out of the Triples i squared them, then i found the sum of their squares. After i took the highest number which was 5 and squared it. As you seen above they equaled the same meaning that they are a Pythagorean Triple.
As stated before i still haven't found a way to predict when Pythagorean Triples will occur but i will continue to do research and look for patterns within the Triples that i do have and can figure out.