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.

Smallest side Middle side Longest side

3 4 5 +1 (from middle side number)

+2 +8

5 12 +4 13 +1 (from middle side number)

+2 +8

7 24 +4 25 +1 (from middle side number)

+2 +8

9 40 +4 41 +1 (from middle side number)

+2 +8

11 60 61 +1 (from middle side number)

a²+b²=c² a²+b²=c² a²+b²=c²

3²+4²=5² 7²+24²=25² 11²+60²=61²

9+16=25 49+576+625 121+3600=3721

25=25 625=625 3721=3721

There was only one pattern I noticed in the smallest side, which was the difference of two between each number.

However, in the middle side the first difference was +8. This then increased by +4. Therefore, the difference between each number was +4.

The longest side patterns were very easy to find. The number was one extra than the number before in the middle side i.e. 4(+1) =5.

Smallest side

N 1 2 3 4 5

Sequence 3 5 7 9 11

st differences +2 +2 +2 +2

2n 2 4 6 8 10

+1 +1 +1 +1 +1

2n+1

Middle side

N 1 2 3 4 5

Sequence 4 12 24 40 60

st differences +8 +12 +16 +20

2nd differences +4 +4 +4

a=4=2

2

2n² 2 8 18 32 50

+2 +4 +6 +8 +10

+2 +2 +2 +2

2n 2 4 6 8 10

+0 +0 +0 +0 +0

2n²+2n

Longest side

N 1 2 3 4 5

Sequence 5 13 25 41 61

st differences +8 +12 +16 +20

Smallest side Middle side Longest side

3 4 5 +1 (from middle side number)

+2 +8

5 12 +4 13 +1 (from middle side number)

+2 +8

7 24 +4 25 +1 (from middle side number)

+2 +8

9 40 +4 41 +1 (from middle side number)

+2 +8

11 60 61 +1 (from middle side number)

a²+b²=c² a²+b²=c² a²+b²=c²

3²+4²=5² 7²+24²=25² 11²+60²=61²

9+16=25 49+576+625 121+3600=3721

25=25 625=625 3721=3721

There was only one pattern I noticed in the smallest side, which was the difference of two between each number.

However, in the middle side the first difference was +8. This then increased by +4. Therefore, the difference between each number was +4.

The longest side patterns were very easy to find. The number was one extra than the number before in the middle side i.e. 4(+1) =5.

Smallest side

N 1 2 3 4 5

Sequence 3 5 7 9 11

st differences +2 +2 +2 +2

2n 2 4 6 8 10

+1 +1 +1 +1 +1

2n+1

Middle side

N 1 2 3 4 5

Sequence 4 12 24 40 60

st differences +8 +12 +16 +20

2nd differences +4 +4 +4

a=4=2

2

2n² 2 8 18 32 50

+2 +4 +6 +8 +10

+2 +2 +2 +2

2n 2 4 6 8 10

+0 +0 +0 +0 +0

2n²+2n

Longest side

N 1 2 3 4 5

Sequence 5 13 25 41 61

st differences +8 +12 +16 +20