I am going to investigate Pythagorean triples where the shortest side is an odd number and all 3 sides are positive integers - I will then investigate other families of Pythagorean triples to see if Pythagoras’ theorem (a²+b²=c²) works

2nd differences +4 +4 +4

a=4=2

2

2n² 2 8 18 32 50

+3 +5 +7 +9 +11

+2 +2 +2 +2

2n 2 4 6 8 10

+1 +1 +1 +1 +1

2n²+2n+1

Smallest side

(2n+1) ²

(2n+1) (2n+1)

2n (2n+1) +1(2n+1)

2nx2n+2nx1+1x2n+1x1

4n²+2n+2n+1

4n²+4n+1

Middle side

(2n²+2n) ²

(2n²+2n) (2n²+2n)

2n² (2n²+2n)+2n (2n²+2n)

2n²x2n²+2n²x2n+2nx2n²+2nx2n

4n4+4n³+4n³+4n²

4n4+8n³+4n²

Longest side

(2n²+2n+1)

(2n²+2n+1) (2n²+2n+1)

2n² (2n²+2n+1) +2n(2n²+2n+1) +1(2n²+2n+1)

4n4+4n³+2n²+4n³+4n²+2n+2n²+2n+1

4n4+8n³+8n²+4n+1

a²+b²=c²

(2n+1) ²+ (2n²+2n) ²=(2n²+2n+1)

4n²+4n+1+4n4+8n³+4n²=4n4+8n³+8n²+4n+1

4n4+8n³+8n²+4n+1=4n4+8n³+8n²+4n+1

To achieve this I took the answers from the smallest, middle and longest sides squared. I then took those answers and converted them into a²+b²=c². I did this to test if Pythagoras' theorem worked. It worked.

X2

Smallest side Middle side Longest side

6 8 10 +2 (from number in middle side)

+4 +16

10 24 +8 26 +2 (from number in middle side)

+4 +24

14 48 +8 50 +2 (from number in middle side)

+4 +32

18 80 +8 82 +2 (from number in middle side)

+4 +40

22 120 122+2 (from number in middle side)

a²+b²=c² a²+b²=c² a²+b²=c²

6²+8²=10² 14²+48²=50² 22²+120²=122²

36+64=100 196+2304=2500 484+14400=14884

00=100 2500=2500 14884=14884

Looking at the table, I have noticed that the difference has increased to +4.

In the middle side, I have noticed that the difference starts with +16 then increases by +8. Therefore, the difference between each number is +8.

The longest side does not have a difference. It is achieved by adding two from the number in the middle side.

Looking at the patterns, they have been achieved by doubling them from the previous table.

Smallest side

N 1 2 3 4 5

Sequence 6 10 14 18 22

st differences +4 +4 +4 +4

4n 4 8 12 16 20

+2 +2 +2 +2 +2

4n+2

Middle side

N 1 2 3 4 5

Sequence 8 24 48 80 120

st differences +16 +24 +32 +40

2nd differences +8 +8 +8

a=8=4

2

4n² 4 16 36 64 100

+4 +8 +12 +16 +20

+4 +4 +4 +4

4n 4 8 12 16 20

+0 +0 +0 +0 +0

4n²+4n

Longest Side

N 1 2 3 4 5

Sequence 10 26 50 82 122

st differences +16 +24 +32 +40

2nd differences +8 +8 +8

a=8=4

2

4n² 4 16 36 64 100

+6 +10 +14 +18 +22

+4 +4 +4 +4

4n 4 8 12 16 20

+2 +2 +2 +2 +2

4n²+4n+2

Smallest side

(4n+2) ²

(4n+2) (4n+2)

4n (4n+2) +2 (4n+2)

4nx4n+4nx2+2x4n+2x2

6n²+8n+8n+4

6n²+16n+4

Middle side

(4n²+4n) ²

(4n²+4n) (4n²+4n)

4n² (4n²+4n) +4n (4n²+4n)

4n²x4n²+4n²x4n+4nx4n²+4nx4n

6n4+16n³+16n³+16n²

6n4+32n³+16n²

Longest side

(4n²+4n+2) ²

(4n²+4n+2) (4n²+4n+2)

4n² (4n²+4n+2) +4n (4n²+4n+2) +2 (4n²+4n+2)

4n²x4n²+4n²x4n+4n²x2+4nx4n²+4nx4n+4nx2+2x4n²+2x4n+2x2

6n4+16n³+8n²+16n³+16n²+8n+8n²+8n+4

6n4+32n³+32n²+16n+4

a²+b²=c²

(4n+2) ²+ (4n²+4n) ²= (4n²+4n+2) ²

6n²+16n+4+16n4+32n³+16n²=16n4+32n³+32n²+16n+4

6n4+32n³+32n²+16n+4=16n4+32n³+32n²+16n+4

To get this I took the answers from squaring the smallest, middle and longest sides. I then transformed these answers into Pythagoras' theory, a²+b²=c², to see if it worked and it did.

X3

Smallest side Middle side Longest side

9 12 15 +1(from number in middle side)

+6 +24

15 36 +12 39 +1(from number in middle side)

+6 +36

21 72 +12 75 +1(from number in middle side)

+6 +48

27 120 +12 123 +1(from number in middle side)

+6 +60

33 180 183 +1(from number in middle side)

I have noticed that the smallest sides' differences have increased to +6.

I have noticed that the middle sides difference. Start with + 24, then increase by +12.

In the longest side, number is achieved by adding one from the number in the middle side.

Another pattern that I noticed was that all the numbers were tripled from the first table.

Prediction

By looking at the first table, the formula was 2n+1. The second table formula was 4n+2. This has been achieved by doubling the first formula. The middle formula in the first table is 2n²+2n. The middle formula for the second table is 4n²+4n. Again, this has been achieved by doubling it. The longest side in the first table is 2n²+2n+1. The second table is if4n²+4n+2.

Therefore, I predict that the smallest sides' formula in the third table would be 6n+3. The middle side will be 6n²+6n.the longest side will be 6n²+6n+3.

Smallest side

N 1 2 3 4 5

Sequence 9 15 21 27 33

st differences +6 +6 +6 +6

6n 6 12 18 24 30

+3 +3 +3 +3 +3

6n+3

Middle side

N 1 2 3 4 5

Sequence 12 36 72 120 180

st differences +24 +36 +48 +60

2nd differences +12 +12 +12

a=12=6

2

6n² 6 24 54 96 150

+6 +12 +18 +24 +30

+6 +6 +6 +6

6n 6 12 18 24 30

+0 +0 +0 +0 +0

6n²+6n

Longest Side

N 1 2 3 4 5

Sequence 15 39 75 123 183

st differences +24 +36 +48 +60

2nd differences +12 +12 +12

a=12=6

2

6n² 6 24 54 96 150

+9 +15 +21 +27 +33

+6 +6 +6 +6

6n 6 12 18 24 30

+3 +3 +3 +3 +3

6n²+6n+3

Smallest side

(6n+3)²

(6n+3) (6n+3)

6n (6n+3) +3 (6n+3)

6nx6n+6nx3+3x6n+3x3

36n²+18n+18n+9

36n²+36n+9

Middle side

(6n²+6n) ²

(6n²+6n) (6n²+6n)

6n² (6n²+6n) +6n (6n²+6n)

6n²x6n²+6n²x6n+6nx6n²+6nx6n

36n4+36n³36n³+36n²

36n4+72n³+36n²

Longest side

(6n²+6n+3) ²

(6n²+6n+3) (6n²+6n+3)

6n² (6n²+6n+3) +6n (6n²+6n+3) +3 (6n²+6n+3)

6n²X6n²+6n²x6n+6n²x3+6nx6n²+6nx6n+6nx3+3x6n²+3x6n+3x3

36n4+36n³+18n²+36n³+36n²+18n+18n²+18n+9

36n4+72n³+72n²+36n+9

a²+b²=c²

(6n+3)²+ (6n²+6n) ²= (6n²+6n+3) ²

36n²+36n+9+36n4+72n³+36n²=36n4+72n³+72n²+36n+9

36n4+72n³+72n²+36n+9=36n4+72n³+72n²+36n+9

Conclusion

I have predicted correctly. The formulas were as I predicted. I also found out that Pythagoras' theorem did work.

Maths Coursework-Mr. Neighbour Jaspreet Athwal 10P