Beyond pythagoras

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The numbers , 4 and 5 satisfy this condition

+ 4 = 5

because = x =

4 = 4 x 4 = 16

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5 = 5 x 5 = 5

and so + 4 = + 16 = 5 = 5

The numbers 5, 1, 1 and 7, 4, 5 also work for this theorem

5 + 1 = 1

because 5 = 5 x 5 = 5

1 = 1 x 1 = 144

1 = 1 x 1 = 16

and so 5 + 1 = 5 + 144 = 16 = 1

7 + 4 = 5

because 7 = 7 x 7 = 4

4 = 4 x 4 = 576

5 = 5 x 5 = 65

and so 7 + 4 = 4 + 576 = 65 = 5 essaybank.co.uk

, 4, 5

Perimeter = + 4 + 5 = 1

Area = ½ x x 4 = 6

5, 1, 1

Perimeter = 5 + 1 + 1 = 0

Area = ½ x 5 x 1 = 0

7, 4, 5

Perimeter = 7 + 4 + 5 = 56

Area = ½ x 7 x 4 = 84

From the first three terms I have noticed the following -

?a? increases by + each term

?a? is equal to the term number times then add 1

the last digit of ?b? is in a pattern 4, , 4

the last digit of ?c? is in a pattern 5, , 5

the square root of (?b? + ?c?) = ?a?

?c? is always +1 to ?b?

?b? increases by +4 each term

(?a? x ?n?) + n = ?b?

Now I will complete the square on ?4n + 6n + ? to see what the solution to this is.

4n + 6n +

4(n +) - +

4(n +) - 7

4(n +) = 7

(n +) = 1.75

n + = 1.875656

n + = -1.875656

n = -1.6771444

n = -4.875656

Arithmatic Progression

I would like to know whether or not the Pythagorean triple ,4,5 is the basis of all triples just some of them.

To find this out I have been to the library and looked at some A-level textbooks and learnt ?Arithmatic Progression?

, 4, 5 is a Pythagorean triple

The pattern is plus one

If a = and d = difference (which is +1) then

= a

4 = a + d

5 = a +d

MQ8FnH from MQ8FnH essay MQ8FnH bank MQ8FnH co MQ8FnH uk

a, a +d, a + d

Therefore if you incorporate this into Pythagoras theorem

a + (a + d) = (a + d)

a + (a + d)(a + d) = (a + d)

a + a + ad + ad + d = (a + d)

a + ad + d = (a + d)

a + ad + d = (a + d)(a + d)

a + ad + d = a + ad + ad + 4d

a + ad + d = 4d + a + 4ad

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If you equate these equations to 0 you get

a - d - ad = 0

Change a to x

x - d - dx = 0

Factorise this equation to get

(x + d)(x - d)

Therefore

x = -d

x = d

x = -d is impossible as you cannot have a negative dimension

a, a+d, a + d

Is the same as

d, 4d, 5d

This tells us that the only Pythagorean triples are , 4, 5 or multiples of , 4, 5 e.g. 6, 8, 10 or 1, 16, 0 etc.

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