BLOG ENTRY #5

SINCE MY LAST BLOG ENTRY I HAVE BEEN INVESTING THE QUESTION: HOW IS THE DIOPHANTS FORMULA RELATED TO THE PYTHAGOREAN THEOREM FORMULA IF THERE ARE ANY RELATIONS AMONG THE TWO AT ALL?
I DID THE SAME THING FOR A DIFFERENT SET OF TRIPLES THAT I DID FOR THE TRIPLE 3-4-5 IN THE PREVIOUS BLOG ENTRY. THE TRIPLES THAT I USED SECOND WAS 16-20-12. WELL I STILL DONT KNO HOW TO GO ABOUT WORKING THE DIOPHANTUS FORMULA BACKWARDS SO I STUCK WITH REARRANGING THE PROCESS. I KEPT THE 1ST STEP:

• 2*4*(2)= 16

THEN INSTEAD I DID STEP 3 SECOND:

• 4^2-2^2= 16-4= 12

THIS WAY I HAD SIDE LENGTHS "a" AND "b">>SIDE LENGTH "a"= 12 AND SIDE LENGTH "b"= 16
I STILL SEEN THE PATTERN THAT I CAN GET THE HYPOTENUSE 2 DIFFERENT WAYS. I COULD DO STEP 2 OF THE DIOPHANTUS FORMULA:

• 2^2+4^2= 4+16= 20

{IF I WAS TO DO THE SECOND PART LAST I REALLY COULD HAVE JUST DONE ALL THE STEPS IN ORDER BECAUSE EITHER WAY I WAS USING THE FORMULA BUT LOOKING AT THE FORMULA IN A DIFFERENT WAY WAS AS USEFUL TO ME AS DOING IT IN THE ORDER DIOPHANTUS WAS QUOTED ON IN MY 1ST BLOG ENTRY.}

OR I COULD HAVE USED THE FORMULA FOR THE PYTHAGOREAN TRIPLE:

• a^2+b^2= c^2

• 12^2+16^2= 20^2

• 144+512= 400

• 656≠ 400

SO IN CONCLUSION MY THEORY OF THE PYTHAGOREAN THEOREM AND THE DIOPHANTUS FORMULA HAVING SOME KIND OF RELATIONSHIP WITH EACH OTHER IS ABSOLUTELY NOT TRUE. IT IS NOT TRUE BECAUSE IT DOESN'T WORK FOR EVERY TRIPLE. THIS MEANS THAT IT ONLY WORKED FOR THE TRIPLE, 3-4-5 BECAUSE IT WAS A COINCIDENCE. THEN AGAIN I ONLY EXPERIMENTED WITH TWO TRIPLES WHICH WERE: 3-4-5 AND 12-16-20 SO IT COULD HAVE ONLY DIDN'T WORK FOR THE 12-16-20 TRIPLE OR IT COULD HAVE ONLY WORKED FOR THE 3-4-5 TRIPLE.

WHAT IS THE REASON THE PYTHAGOREAN THEOREM AND THE DIOPHANTUS FORMULA DON'T MIX OR CANT WORK TOGETHER TO GET TRIPLES? WHAT IS THE MAIN PROBLEM OR GLITCH THAT WONT ALLOW THIS TO WORK? IS THERE ANY WAY TO MAKE IT WORK? FOR EXAMPLE CAN YOU GO AROUND THE PROBLEM AND MAKE A SOLUTION TO IT?

BLOG ENTRY #4

Im investigating the question WHILE MEASURING A TRIANGLE WHEN CAN YOU TELL THAT THE LENGTHS ARE A SET OF PYTHAGOREAN TRIPLES? IF SO IS HIS A MATH THING THAT YOU ARE JUST GOING TO HAVE TO REMEMBER ALL OF THEM OR IS THERE A WAY YOU CAN FIGURE IT OUT USING OTHER METHODS? like for example can you use the diaphantus formula to prove that its one of many Pythagorean Triples?
I STARTED INVESTIGATING BY LOOKING AT A TRIANGLE THAT HAD THE SIDE LENGTHS 3-4-5 WHICH IS AN PYTHAGOREAN TRIPLE. TO GO ABOUT INVESTIGATING MY QUESTION ALONG WITH THIS TRIANGLE I WENT BACK TO THE DIOPHANTUS FORMULA. WHEN I USED THE FORMULA TO FIND THIS TRIPLE MY CALCULATIONS WERE:
1*2*(2)= 4
1^2+1^2= 1+4= 5
2^2-1^2= 4-1= 3
IF I DIDNT KNOW THAT 3-4-5 WAS A TRIPLE AND ALL I WAS GIVEN WAS TWO LENGTHS HOW WOULD I KNO THAT THIS WOULD B A TRIPLE?
WELL THATS WHAT I WANT TO KNO ND BEEN INVESTIGATING AND HAVEN'T REALLY GOTTEN ANY WHERE. IF I WERE GIVEN SIDES "A" AND "B" WHICH IS 3 AND 4 HOW CAN I FIND OUT THE HYPOTENUSE? WELL THAT WAS A QUESTION THAT WAS EASY FOR ME TO ANSWER BUT THATS WHERE I BEGAN. THE FORMULA THAT WILL HELP ME WITH THAT ANSWER WAS THE PYTHAGOREAN THEREOM WHICH IS:
a^2+b^2= c^2
3^2+4^2= 5^2
SEE ALREADY KNEW THIS SO I THOUGHT SOME MORE. I CAME UP WITH THIS IDEA AS TO IF IT IS POSSIBLE TO WORK THE DIOPHANTUS FORMULA BACKWARDS. WELL I DIDNT KNO HOW TO GO ABOUT DOING THAT SO I REARRANGED THE PROCESS. I KEPT THE 1ST STEP:

• 1*2*(2)= 4

THEN INSTEAD I DID STEP 3 SECOND:

• 2^2-1^2= 4-1= 3

THIS WAY I HAD SIDE LENGTHS "a" AND "b"
I THOUGHT ABOUT IT AND SAID I CAN GET THE HYPOTENUSE 2 DIFFERENT WAYS. I COULD HAVE DID STEP 2 OF THE DIOPHANTUS FORMULA:

• 1^2+1^2= 1+4= 5

OR I COULD HAVE USED THE FORMULA FOR THE PYTHAGOREAN TRIPLE:

• a^2+b^2= c^2

• 3^2+4^2= 5^2

I NOTICED THAT EITHER FORMULA WOULD HAVE HELPED ME GET THE RESULTS THAT I WANTED.
BUT THIS DID LEAVE ME WITH ANOTHER QUESTION IN MY MIND. HOW IS THE DIOPHANTS FORMULA RELATED TO THE PYTHAGOREAN THEROEM FORMULA IF THERE ARE ANY RELATIONS AMOUNG THE TWO AT ALL?

BLOG ENTRY#3

the question i began with was:Pythagorean triples are right triangles that have integer side lengths. how many of these are there? can you predict when they will occur?
in the second blog entry i began noticing patterns about the triples that i had discovered and listed in my previous entries. one question that was brought to my attention was if it was possible to have more than one triple that started with the same number? the answer is yes. in the data that i collected so far i have found two pair of triples that begin with the same number.
TRIPLES THAT START w/ THE SAME #:

9-12-15;	12-13-16; 20-21-19
9-40-41;	12-16-20; 20-21-29

one question that im going to do my best to investigate is ,WHILE MEASURING A TRIANGLE WHEN CAN YOU TELL THAT THE LENGTHS ARE A SET OF PYTHAGOREAN TRIPLES? IF SO IS HIS A MATH THING THAT YOU ARE JUST GOING TO HAVE TO REMEMBER ALL OF THEM OR IS THERE A WAY YOU CAN FIGURE IT OUT USING OTHER METHODS? like for example can you use the diaphantus formula to prove that its one of many Pythagorean Triples?

BLOG ENTRY #2

START w/ Odd #s	START w/ EVEN #	ALL EVEN #s	EVEN Odd Odd
3-4-5	12-13-16	6-8-10	6-15-17
5-12-13		10-24-26	20-21-29
7-24-25		16-30-34	16-63-65
9-12-15		12-16-20	72-135-153
9-40-41			24-143-145

3-4-5	3-4-5	6-8-10	8-15-17	5-12-13
3(2)-4(2)-5(2)	3(3)-4(3)-5(3)	6(2)-8(2)-10(2)	8(2)-15(2)-17(2)	5(2)-12(2)-13(2)
6-8-10	9-12-15	12-16-20	16-30-34	10-24-26

I noticed that it is rare to have a triple with the pattern of even-odd-even, but there are a lot of triples with the pattern of odd-even-odd.
I also noticed that in the triples 3-4-5 & 9-12-15, in the first group, if you multiplied all the numbers by 3 you would get the numbers of the second group of Triples.
There is another pattern similar with two Triples in the all even column (shown below). if you multiply all the numbers in 6-8-10 by 2 you will get 12-16-20 which is also a Pythagorean Triple.
Another thing that I observed was about the Pythagorean Triple 8-15-17. if you multiply the numbers by 2 the product is 16-30-34 which is another Pythagorean Triple that I found using the formula of Diophantus.
When you multiply the numbers in the Pythagorean Triple, 5-12-13 by 2 then the product is the Triple, 10-24-26.
One thing that stood out to me was that the Triples that involve odd numbers. When I multiplied them by 2 they turned out to be Triples that have all even numbers in them, that happened with three sets of Pythagorean Triples which were, 3-4-5, 8-15-17, and 5-12-13.

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.