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# Proving a2 + b2 = c2 Using Odd Numbers

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Introduction

Proving a2 + b2 = c2 Using Odd Numbers

a2 + b2 = c2

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

4n2 + 8n3 + 4n2 + 4n + 1 = 4n4 + 4n3 + 2n2 + 4n3 + 4n2 +2n + 2n2 +2n +1

4n4 + 8n3 + 8n2 + 4n + 1 = 4n4 +8n3 + 8n2 +4n + 1

The above proves that a2 + b2 = c2 is correct using odd numbers and is a overall formula for Pythagorean triples.

PerimeterOdds

Perimeter = a + b + c

a= 2n + 1

b= 2n2 + 2n

c= 2n2 + 2n + 1

= 4n2

Middle

= 18 + 30 + 12

= 60

Follow n=3 across on the evens table the perimeter says 60 my formulae is correct.

Odds Side c

c = b + 1

c = 2n2 +2n + 1

So from looking at the table and extending it, I have managed to come up with the conclusion that:

ODDS

a = 2n + 1

b = 2n2 + 2n

c = 2n2 + 2n + 1

Evens Table

 Length of (a)

Conclusion

n + 2 x a = b + c also I have noticed, if you look at c starting down from 10 it is 3x6, 4x8, 5x10 ect.

Area - Odds

Area- a x b

2

(2n+1) (2n2 + 2n)

2

= 4n3 + 4n2 + 2n2 + 2n

2

= 4n3 + 6n2 + 2n

2

= 2n3 + 3n2 + n

To prove this is correct I will take n=3:

=2 x 32 + 3 x 32 + 3

=54 + 27 + 3

= 84

Follow n=3 across to area on the odds table and its 84 this proves my formula is correct.

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