1. Pythagorean Triples – Part 2
Slideshow 39, Mathematics
Mr. Richard Sasaki, Room 307

2. Objectives
Check some points that we could have learned from the given project (relationships & rules with triples)
Learn and use some laws in triples
Introduce triangular numbers

3. Rules and Patterns for Triples
For 𝒂= 𝒎 𝟐 − 𝒏 𝟐 , 𝒃=𝟐𝒎𝒏, 𝒄= 𝒎 𝟐 + 𝒏 𝟐 , we can insert any values of 𝑚 and 𝑛 where
𝑚>𝑛
Triples always consist of three positive integers where they are either…
All even numbers (always non-primitive)
Two odd numbers and an even number
Some properties for primitive triples include…
Either 𝑎 or 𝑏 is odd, 𝑐 is odd
One of 𝑎 or 𝑏 has a factor of 3 and 4
(It could be that either 𝑎 or 𝑏 has both factors or 𝑎 has one and 𝑏 has one.)
Either 𝑎, 𝑏 or 𝑐 has a factor of 5
At maximum, one of 𝑎, 𝑏 or 𝑐 is square

4. Rules and Patterns for Triples
An interesting one is if you calculate 𝑐−𝑎 𝑐−𝑏 2 where 𝑎, 𝑏,𝑐∈ ℤ + , you get a square number if it’s valid.
Example
Using 𝑐−𝑎 𝑐−𝑏 2 , test whether (33, 56, 65) may be a valid triple.
65−33 65−56 2 =
288 2 =
144
The test is passed as 144 is a square number.
Note: This formula can produce square numbers for invalid triples. (3, 4, 12) would give us 36 which is square but it is not a valid triple. However, if it is not square, it is definitely invalid.

5. Rules and Patterns for Triples
If you enter any value for 𝑎 for a triple (𝑎, 𝑏, 𝑐) where 𝒂<𝒃<𝒄, and 𝒂 is a prime number other than 2, 𝑏 and 𝑐 can be calculated.
In other words, triples exist for ∀ 𝑎 where 𝑎>2 and prime. In fact they will be primitive.
So how do we calculate them?
Example
Consider a triangle with its shortest leg 13. Find the lengths of the other two edges assuming all edges hold integer values.
You may have found that when 𝑎 is prime, for the triple (𝑎, 𝑏, 𝑐), 𝑐=𝑏+1. 13
(𝑏 and 𝑐 are consecutive integers.)

6. Rules and Patterns for Triples
Using the fact that 𝑐=𝑏+1 and we are given 𝑎, we can make a formula to calculate 𝑏 and 𝑐. 85 13 84
As always, start with 𝑎 2 + 𝑏 2 = 𝑐 2 . Using 𝑐=𝑏+1, we can write a simpler equation about 𝑎, 𝑏 and 𝑐.
𝑎 2 + 𝑏 2 = 𝑐 2
Note: 𝑎 2 =𝑏+𝑐 is just for looking of pairs that fit 𝑏 and 𝑐, not for calculation.
𝑎 2 = 𝑐 2 − 𝑏 2
This is in terms of 𝑏.
𝑎 2 = (𝑏+1) 2 − 𝑏 2
𝑎 2 = 𝑏 2 +2𝑏+1− 𝑏 2
𝒂 𝟐 =𝟐𝒃+𝟏
𝑎 2 =𝑏+(𝑏+1)
13 2 =𝑏+𝑐
This is in terms of 𝑐.
𝒂 𝟐 =𝒃+𝒄
169=𝑏+𝑐
𝑎 2 =(𝑐−1)+𝑐
169=84+85
𝒂 𝟐 =𝟐𝒄−𝟏
𝑏=84, 𝑐=85

7. 7−3 7−6 2 =2 (not a square number)
None of these numbers have a factor of 4 or 3.
(3, 4, 5)
(7, 24, 25)
(11, 60, 61)
(13, 84, 85)
(17, 144, 145)
(19, 180, 181)
(23, 264, 265)
(29, 420, 421)
(9, 40, 41)
Doesn’t work
Doesn’t work
(15,112, 113)
𝑎 must be odd

8. When 𝑎 isn’t prime When 𝑎 isn’t prime, 𝑎 isn’t unique.
For example, when 𝑎=9, we would produce (9, 40, 41) which is primitive but also (9, 12, 15) exists (derived from (3, 4, 5)).
So for any integer 𝑎, we can divide by whatever number to make it prime (and not 2), use the formula 𝑎=𝑏+𝑐 to produce some (𝑎, 𝑏, 𝑐) and then multiply the number back in.
So, if 𝑎=20, we divide by 4 to get 𝑎=5⇒
(5, 12, 13)
and then multiply 4 back to get
(20, 48, 52)
Also, (20, 21, 29) exists.
Did you find another triple for 𝑎=20?

9. Multiples of 4 You may have found for 𝑎=20. (20, 99, 101)
For all values of 𝑎 where 𝑎 is a multiple of 4, a primitive triple exists where 𝑏 and 𝑐 are consecutive odd numbers.
Let’s make a formula we can use!
101
For 𝑎 2 + 𝑏 2 = 𝑐 2 where 𝑐=𝑏+2… 20
𝑎 2 + 𝑏 2 = 𝑏+2 2 99
𝑎 2 + 𝑏 2 = 𝑏 2 +4𝑏+4
𝑎 2 =4𝑏+4
=𝑏+𝑐
𝑎 2 =2𝑏+(2𝑏+4)
𝑎 2 =2𝑏+2(𝑏+2)
200=𝑏+𝑐
𝑎 2 =2𝑏+2𝑐
200=99+101
𝒂 𝟐 𝟐 =𝒃+𝒄
So (20, 99, 101) is true.

10. (8, 15, 17)
(12, 35, 37)
(24, 143, 145)
(28, 195, 197)
We get (4, 3, 5) which is still valid and primitive but 𝑎 is not the lowest value.
(5, 12, 13)
(16, 63, 65)
(15, 112, 113)
(32, 255, 257)
(31, 480, 481)
(21, 220, 221)
760
785
Other even numbers are fine too. 𝑎=10 gives us 10, 24, 26 . This is valid but not primitive.

11. Triangular Numbers You learned about these in the Winter Homework.
They follow the list…
1, 3, 6, 10, 15, 21, …
As you know, they are written in the form 𝑇 𝑛 where 𝑛 is the number’s position, or the number of dots on an edge.
What do these numbers have to do with Pythagorean triples?

12. Triangular Numbers
If you checked the grading sheet (as advised), you would have seen…
Your project was for 1≤𝑚≤10, 1≤𝑛≤10, right? How many triples were there? 45
For this test, 𝑥=10.
For 1≤𝑚≤9, 1≤𝑛≤9, you’d simply ignore results where 𝑚=10 or 𝑛=10. How many are there? (𝑥=9) 36
In fact, as you decrease the values of 𝑥, you’ll see…
Value of 𝒙 1 2 3 4 5 6 7 8 9 10
# of Triples 𝟑𝟔 𝟒𝟓 𝟎 𝟏 𝟑 𝟔 𝟏𝟎 𝟏𝟓 𝟐𝟏 𝟐𝟖
This produces the triangle numbers!

13. Triangular Numbers So for 1≤𝑚, 𝑛≤𝑥, is the number of valid triples.
𝑇 𝑥−1
If you were asked to calculate 𝑇 50 , how would you?
= 𝑛=1 𝑥 𝑛 =
𝑥 𝑥+1 2
𝑇 𝑥 =
1+2+…+𝑥
Using 𝑥 𝑥 is the easiest way to calculate this. =
𝑇 50 =
=1275
Let’s prove 𝑇 𝑥 = 𝑥 𝑥
Consider 𝑇 𝑥 =1+2+…+𝑥 and 𝑇 𝑥 =𝑥+ 𝑥−1 +…+1.
∴ 2 𝑇 𝑥 = 1+𝑥 + 2+ 𝑥−1 +…+ 𝑥+1 .
All terms can be written as (𝑥+1) and there are 𝑥 terms.
∴2 𝑇 𝑥 =𝑥 𝑥+1 ⇒ 𝑇 𝑥 = 𝑥 𝑥+1 2

14. 10
153
276
𝑇 𝑛−1
𝑛 2 𝑛
𝑛+1 𝐶 2 = 𝑛+1 ! 2 𝑛−1 !
= 𝑛+1 ∙𝑛∙ 𝑛−1 ∙…∙1 2(𝑛−1)∙…∙1
= 𝑛+1 ∙𝑛 2 = 𝑛 𝑛+1 2