Square Number Theorem 4ab=(a+b)²-(a-b)²

[M_B_S]

Square Number Theorem 4ab=(a+b)²-(a-b)²

n,a,b element N n=ab =>

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4(ab)=(a+b)²-(a-b)²

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Every 4(ab) and odd number (ab) is the difference of two perfect squares.

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Every square number is the sum of two square numbers while 0² is a square number or (ab) is square number

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Every number is the difference of two squares .

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q.e.d.

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correction

 

[HallsofIvy]

n,a,b element N n=ab =>

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4(ab)=(a+b)²-(a-b)²

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Every 4(ab) and odd number (ab) is the difference of two perfect squares.

Yes, if ab is odd then both a and b are odd so a- b and a+ b are even. \(\displaystyle x^2- y^2= (x- y)(x+ y)= ab then we can take a= x-y and b= x+y (with b< a). So 2x= (x+y)+(x- y)= a+ b and x= (a+ b)/2, 2y= x+y-(x-y)= b- a and y= (b- a)/2.

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Every number is the sum of two square numbers while 0² is a square number or (ab) is square number

How is 3 the sum of two squares?[.quote]

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Every number is the difference of two squares .

You said above that every odd number is the sum of difference of two squares. But you have not shown that every even numbers is a difference of two squares.

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q.e.d.

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Every

Is something missing here?\)

 

[M_B_S]

See my correction:???: sorry for that. Every number n is the difference of two squares in N,Q,C,R

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4n=(n+1)²-(n-1)²

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Remember a square divided by four is still a square

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[lookagain]

See my correction:???: sorry for that. Every number n is the difference of two squares in N,Q,C,R

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4n=(n+1)²-(n-1)²

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> > > Remember a square divided by four is still a square < <

 

[M_B_S]

> > > Remember a square divided by four is still a square < < <[/QUOTE]

a²/4 =(a/2)²

 

[M_B_S]

Simple Proof: Every odd number is the difference of two perfect square numbers

n in N

every odd number is a difference of two square numbers

2n+1=(n+1)²-n²

 

[M_B_S]

Any square number (a²+b²)² is the sum of two perfect squares

Proof: (2ab)²+(a²-b²)²=(a²+b²)²

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q.e.d.

 

[M_B_S]

Every even perfect square number is the difference of two perfect square numbers

Proof

a,b,n in N ; n=ab


(2*a*b)²=(a²+b²)²-(a²-b²)²

q.e.d.

 

[M_B_S]

Took you 6 months?

No, look 1 post above

But for the kids i put it in.

CU

 

[M_B_S]

What does a proof fully proof?

....so took you 6 months to put it in

Hi Denis

Yes my kids ask me one week ago if every perfect squared number odd or even is the difference of two squares?

Answer: Yes it is.

 

[M_B_S]

Squared numbers are the difference of two squared numbers

WHAT do you mean: "my kids"?
Are you a teacher?

Both teacher and father

All perfect square numbers are the difference of two perfect square numbers

 

[Quaid]

Are you including 0 as a perfect square?

Zero is a perfect square. (A perfect square is a number that can be expressed as the product of two equal integers.)

Have you seen a situation where zero was not included in the set of perfect squares?

 

[lookagain]

Not listed here:
http://www.mathwarehouse.com/arithmetic/numbers/list-of-perfect-squares.php

Well, their own list is incomplete, even by their own definition.

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Related Links: What is a perfect Square? <--------- Click on their link above their list of perfect numbers and see this:


"A perfect square is a number that can be expressed as the product of two equal integers."


That includes 0.

 

[Quaid]

Not listed [at mathwarehouse]

That's a shame. Did you click on this link at that page? What is a perfect Square?

And if 0 is used here, then why is OP stating this:
"All perfect square numbers are the difference of two perfect square numbers"

n^2 - 0^2 = n^2

The statement does not claim that the difference of two perfect square numbers is a perfect square number.

It claims that a perfect square number may be expressed as the difference of two perfect square numbers.

0 = 0 - 0

 

[M_B_S]

0 is a square number

http://mathworld.wolfram.com/SquareNumber.html

But to make it easy

a odd number squared is a odd number => proof => difference of two perfect squared numbers

a even number squared is a 4n number => proof => difference of two perfect squared numbers

q.e.d.

 

[M_B_S]

Another Four Square Theorem

Out of this follows that every n in N with 0 can be expressed as


n= (a²+b²)-(c²+d²) <=> (a²-c²)+(b²-d²) ; n,a,b,c,d, out of Natural Numbers with 0

Proof:

Every even number is the sum of two odd numbers

Every odd number is the sum of an odd number plus an even number 0 modolu 4

q.e.d.

 

[M_B_S]

Proof Prime Number Product (c+d)(c-d) = c²-d²

So let a and b different odd prime numbers.

=> a*b = c²-d² out of square number theorem because its an odd number

=> c²-d² = (c+d)(c-d) => a=(c+d) and b=(c-d) because its binominal

=> If (c+d) is prime there exist allways a prime (c-d)

Examples:

7*3 = 5²-2² = (5+2)(5-2) = 21

11*5 = 8²-3² = (8+3)(8-3) = 55

Thats very hot stuff because:

(c+d) + ( c-d) = 2c The famous Goldbach Conjectur is true because 2c is every even number ?


M_B_S

 

[M_B_S]

Every n^i is a difference between a²-b² , i>1

Proof


n^i with n,i,a,b,c in N , i>1

=> n^i = a²-b²

A) If n is odd => n^i is odd.

B) If n is even => n^i is a 4(c) number

Examples:
7³ = 28²-21²
2^256 = 2²*2²*2^252= (2²+2^252)²-(2^252-2²)²