# 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&#146; theorem (a&sup2;+b&sup2;=c&sup2;) works

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Introduction

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 (a�+b�=c�) 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) a�+b�=c� a�+b�=c� a�+b�=c� 3�+4�=5� 7�+24�=25� 11�+60�=61� 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 1st differences +2 +2 +2 +2 2n 2 4 ...read more.

Middle

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 1st 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 1st differences +16 +24 +32 +40 2nd differences +8 +8 +8 a=8=4 2 4n� 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 4n�+4n Longest Side N 1 2 3 4 5 Sequence 10 26 50 82 122 1st differences +16 +24 +32 +40 2nd differences +8 +8 +8 a=8=4 2 4n� 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 4n�+4n+2 Smallest side (4n+2) ...read more.

Conclusion

Smallest side N 1 2 3 4 5 Sequence 9 15 21 27 33 1st 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 1st differences +24 +36 +48 +60 2nd differences +12 +12 +12 a=12=6 2 6n� 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 6n�+6n Longest Side N 1 2 3 4 5 Sequence 15 39 75 123 183 1st differences +24 +36 +48 +60 2nd differences +12 +12 +12 a=12=6 2 6n� 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 6n�+6n+3 Smallest side (6n+3)� (6n+3) (6n+3) 6n (6n+3) +3 (6n+3) 6nx6n+6nx3+3x6n+3x3 36n�+18n+18n+9 36n�+36n+9 Middle side (6n�+6n) � (6n�+6n) (6n�+6n) 6n� (6n�+6n) +6n (6n�+6n) 6n�x6n�+6n�x6n+6nx6n�+6nx6n 36n4+36n�36n�+36n� 36n4+72n�+36n� Longest side (6n�+6n+3) � (6n�+6n+3) (6n�+6n+3) 6n� (6n�+6n+3) +6n (6n�+6n+3) +3 (6n�+6n+3) 6n�X6n�+6n�x6n+6n�x3+6nx6n�+6nx6n+6nx3+3x6n�+3x6n+3x3 36n4+36n�+18n�+36n�+36n�+18n+18n�+18n+9 36n4+72n�+72n�+36n+9 a�+b�=c� (6n+3)�+ (6n�+6n) �= (6n�+6n+3) � 36n�+36n+9+36n4+72n�+36n�=36n4+72n�+72n�+36n+9 36n4+72n�+72n�+36n+9=36n4+72n�+72n�+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 ...read more.

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