Difference 3 5 7 9 11 13
Difference in differences 2 2 2 2 2
Difference 3 5 7 9 11 13
+ 1
Value of 2n 2 4 6 8 10 12
So the formula is:
2n² + 2n + 1
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 25.
2n² + 2n + 1
2(3²) + (2 x 3) + 1
18 + 6 + 1
= 25
It is correct!
We should note that the formula for the longest number is identical to the formula of the middle number except that the larger number adds 1 to the formula, which also ties in with the observation I made earlier about the middle and largest numbers being consecutive.
We have succeeded in using formulas to work out, odd starting Pythagorean triples using ‘n’.
We must now look at the rest of the task, by which I mean the area and perimeter, parts of the table. Firstly we shall look at the perimeter:
12 30 56 90 132 182
18 26 34 42 50 1st difference
8 8 8 8 2nd difference
The 2nd difference is 8 so we must find the formula for the ‘perimeter’.
8 / 2 = 4 [8 is halved due to the use of n²
So we must look at 4n²
Sequence 12 30 56 90 132 182
Value of 4n² 4 16 36 64 100 144
Difference 8 14 20 26 32 38
Difference in differences 6 6 6 6 6
Difference 8 14 20 26 32 38
+2
Value of 6n 6 12 18 24 30 36
So the formula is:
4n² + 6n + 2
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 56.
4n² + 6n + 2
(4 x 9) + (6 x 3) +2
36 + 18 + 2
= 56
It is correct!
We can now look at the area to complete the table of formulas,
6 30 84 180 330 546
24 56 96 150 216 1st difference
30 42 54 76 2nd difference
12 12 12 3rd difference
This is hard to do due to the 3rd difference rather than the first or second differences in the previous formulas, so I had to teach myself how to 3rd differences with the use of ‘n’.
I will not worry about the how to get this formula in this way, but first a shall use the original formula to work out area with the use of the other numbers, small number x middle number x a half, I will use this to work out, simply my formula. I will use the formulas for the small number and multiply it by the middle number formula then divide it by 2.
Formula for small number x Formula for middle number x a half
2n + 1 x 2n² + 2n x 0.5
I will work out the first part of multiplying the formulas together:
2n x 2n² = 4n³
(2n + 1)(2n² + 2n)
1 x 2n = 2n
so far we have 4n³ + 2n but we have to complete the multiplying out of the brackets:
2n x 2n = 4n²
(2n + 1)(2n² + 2n)
1 x 2n² = 2n²
We now have 4n³ + 2n + 4n² + 2n² which simplified is:
4n³ + 6n² + 2n
but it doesn’t finish yet because we have to half it to keep with the formula to work out the area using the middle and small numbers.
(4n³ + 6n² + 2n) = 2n³ + 3n² + n
2
This is the formula for the area, which will help the investigation but also help me personally to be able to work out how to use n³.
From my results I can conclude that with the use n³ you must divide the 3rd difference by 6 I will show the process:
6 30 84 180 330 546
24 56 96 150 216 1st difference
30 42 54 76 2nd difference
12 12 12 3rd difference
So we must look at 2n³, because you must divide by 6
Sequence 6 30 84 180 330 546
Value of 2n³ 2 16 54 128 250 432
Difference 4 14 30 52 80 114
Difference2 10 16 22 28 34
Difference3 6 6 6 6
So we must use 3n² also because of the use of n² I had to half the difference3
Difference 4 14 30 52 80 114
Value of 3n² 3 12 27 48 75 108
+ n
Difference4 1 2 3 4 5 6
So the formula is:
2n³ + 3n² + n
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 84 .
2n³ + 3n² + n
(2 x 3³) + (3 x 3²) + 3
54 + 27 + 3
= 84
It is correct!
To recap on all the formulae;
Short side 2n + 1
Middle side 2n² + 2n
Long side 2n² + 2n + 1
Perimeter 4n² + 6n + 2
Area 2n³ + 3n² + n
From this you can work out extra formulae within the table, an example:
Area = perimeter x n x ½
4n² + 6n + 2 [x n
4n³ + 6n² + 2n [x ½
= 2n³ + 3n² + n = area’s formula.
Another formula that can be seen in the recap is that to get the middle side you can do something to the short side, which is to multiply the short side by n then add n.
2n + 1 [x n
= 2n² + n [+ n
= 2n² + 2n
= middle number formula.
This can also be used to find the largest number by simply adding 1.
2n² + 2n [+ 1
2n² + 2n + 1
All these formulae can be reversed to get the opposite conclusion.
The Pythagorean triples are not just numbers they can be into right-angled triangles so to look for more patterns we will look at the angles and gradients of the Pythagorean triples.
(not to scale)
Gradient: 4/3 = 1⅓
6
3
4
Gradient: 12/5 = 2.4
13
5
12
Gradient: 24/6= 3.43 (1dp)
25
7
24
41 Gradient: 40/9 = 4.44
9
40
The pattern I first came across was that the top angle was increasing in size while the bottom angle was decreasing.
The top angle: 53˚ 67˚ 73˚ 77˚
+ 14 + 6 + 3
The bottom angle: 37˚ 23˚ 17˚ 13˚
-14 -6 -3
The bottom angle is decrease at the same rate as the top is increasing, this is because triangles have to have a total angle of 180˚ and because 90˚ has been taken due to the fact that it is a right-angled triangle the rest must add to 90˚ and these angles do but must change accordingly so they stay at 90˚.
The top angle is increased by 14 then is increased by around half of 14, (this is only true if you give around a 1˚ of error in the measurement of the angle) 6 then this is the next number in the sequence is increased by a half of 6, 3. So the angle may keep increasing this way. This is also true for the bottom angle but it is decreased, not increased.
The gradient: 1⅓ 2.4 3.43 4.44
+ 1 .07 + 1.01 +1.01
This gradient is around 1.01 added to it in the sequence
Even Pythagorean triples:
To get an even Pythagorean triple is quite simple because Pythagorean triples can be multiplied and divided to make different numbers, as long as it is to all the sides. This is a form of enlargement.
All the Pythagoreans triples seen in the first table, multiplied by 2:
The formulae I have seen in the previous odd starting Pythagorean triples have been times by 2, apart from the area which has a different formula all together, I will look into this later in the investigation.
Even short side:
6 10 14 18 22 26
4 4 4 4 4 1st difference
The 1st difference is 4 so we can find the formula for the ‘smallest number’.
So we must look at 4n
Sequence 6 10 14 18 22 26
Value of 4n 4 8 12 16 20 24
Difference 2 2 2 2 2 2
So the formula is:
4n + 2
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 14 .
4n + 2
4 x 3 + 2
12 + 2
= 14
It is correct!
The formula has been multiplied by 2, from the original, and works for the even short side. This is only for the even numbers in the table above, because the original Pythagorean triples were multiplied by 2. So the formula for the originals must be accordingly changed using the formula, for the short side, and multiplying by ‘x’, x(2n + 1) will work out any short side of a Pythagorean triple if is a multiple of the original odd Pythagorean triples. x is the number you multiply the originals to get the other Pythagorean triples.
To see if there are any more formulae in the table we will first look at the ‘even middle number’ sequence:
8 24 48 80 120 168
16 24 32 40 48 1st difference
8 8 8 8 2nd difference
The 2nd difference is 8 so we must find the formula for the ‘even middle number’.
8 / 2 = 4 [8 is halved due to the use of n²
So we must look at 4n²
Sequence 8 24 48 80 120 168
Value of 4n² 4 16 36 64 100 144
Difference 4 8 12 16 20 24
Difference in difference 4 4 4 4 4
Difference 4 8 12 16 20 24 0
Value of 4n 4 8 12 16 20 24
So the formula is:
4n² + 4n
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 48.
4n² + 4n
4(3²) + (4 x 3)
36 + 12
= 48
It is correct!
This is also an example of the conclusion I drew earlier where the formula is multiplied by the same number that the original Pythagorean triples were. The original formula was 2n² + 2n this formula is twice that.
We will now look at the even long side and look for a pattern, and see if the same conclusion is drawn:
10 26 50 82 122 170
16 24 32 40 48 1st difference
8 8 8 8 2nd difference
The 2nd difference is 8 so we must find the formula for the ‘even largest number’.
8 / 2 = 4 [8 is halved due to the use of n²
So we must look at 4n²
Sequence 10 26 50 82 122 170
Value of 4n² 4 16 36 64 100 144
Difference 6 10 14 18 22 26
Difference in differences 4 4 4 4 4
Difference 6 10 14 18 22 26 + 2
Value of 4n 4 8 12 16 20 24
So the formula is:
4n² + 4n + 2
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 50.
4n² + 4n + 2
4(3²) + (4 x 3) + 2
36 + 12 + 2
= 50
It is correct!
This also follows the conclusions above.
We have succeeded in using formulas to work out, Pythagorean triples using ‘n’.
We must now look at the rest of the task, by which I mean the area and perimeter, parts of the table. Firstly we shall look at the perimeter:
24 60 112 180 264 364
36 52 68 84 100 1st difference
16 16 16 16 2nd difference
The 2nd difference is 16 so we must find the formula for the ‘even perimeter’.
16 / 2 = 8 [16 is halved due to the use of n²
So we must look at 4n²
Sequence 24 60 112 180 264 364
Value of 8n² 8 32 72 128 200 288
Difference 16 28 40 52 64 76
Difference in differences 12 12 12 12 12
Difference 16 28 40 52 64 76
+ 4
Value of 12n 12 24 36 48 60 72
So the formula is:
8n² + 12n + 4
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 112.
8n² + 12n + 4
(8 x 9) + (12 x 3) + 4
72 + 36 + 4
= 112
It is correct!
This also follows the pervious patterns of multiplying by 2 the formulae in the original odd Pythagorean triples.
We will know look at the area of the even table of Pythagorean triples, similar to enlargement the area doesn’t follow the same pattern of multiplying the area by what you enlarged it by, so it will require a more tougher research.
24 120 336 720 1320 2184
96 216 384 600 864 1st difference
120 168 216 264 2nd difference
48 48 48 3rd difference
So we must look at 8n³, because you must divide by 6
Sequence 24 120 336 720 1320 2184
Value of 8n³ 8 64 216 512 1000 1728
Difference 16 56 120 208 320 456
Difference2 40 64 88 112 136
Difference3 24 24 24 24
So we must use 12n² also because of the use of n² I had to half the difference3
Difference 16 56 120 208 320 456
Value of 12n² 12 48 108 192 300 432
+ 4n
Difference4 4 8 12 16 20 24
So the formula is:
8n³ + 12n² + 4n
To check that the formula is correct we can try and put it into the sequence:
n = 3 the sequence number for this is 336 .
8n³ + 12n² + 4n
(8 x 3³) + (12 x 3²) + (4 x 3)
216 + 108 + 12
= 336
It is correct!
This shows that the area works in a different way to the others, the others are multiplied by the number that the original Pythagorean triples were multiplied by, while the area needs to be multiplied by 4, this shows that the other areas of Pythagorean triples formulae can be worked out by multiplying the original Pythagorean triples multiplier by 2 to find the formula for the areas.
We will check my theory is correct by looking at the originals multiplied by 4
I have checked and found that my theory was correct due to the differences in the table.
The final topic I will mention is a circle drawn to join all of the interior sides of a triangle and show you the relations between the length of sides of the triangle and the radius of the circle.
The “r” stands for radius of the circle inscribed in the triangle, “a” stands for the short side of the right-angled triangle and “b” stands for the medium side. I will now let you think about the diagrams for a little bit and see if you can see the relationships between the radius and sides.
Note: this can also prove that a² + b² = c²
The conclusion:
My conclusion for the task set is that Pythagorean triples can be worked out by the use of the first odd numbered Pythagorean triples (3 – 13) these can be multiplied to any Pythagorean triple and there for the formulae must be changed, to the other Pythagorean triples needs by multiplying formulae by the multiple of the original triples apart from the area which needs to be multiplied again by 2 so the different measurement (units³) can be accounted for. I have also noticed that the shortest and medium sides of the triangle are made up of the radius of an interior circle, either by the medium side minus the radius to for the shortest side or the shortest side minus the radius added to the medium side minus the radius to form the longest side.
The investigation has shown that Pythagorean triples can be predicted but there may be some that I have not accounted for in my investigation.
-Thank You For Reading
Lok Man Lee
