> On 2011-08-08, Deep <email@example.com> wrote: > > I would appreciate it very much if someone can refer me to an > > appropriate reference where following statement is justified. > > > > Statement: Every integer N that is not 2mod4 can be expressed as the > > difference of two squares. > > No reference necessary. > > For odd numbers: 2n+1 = (n+1)^2 - n^2. > For numbers divisible by 4: 4n = (n+1)^2 - (n-1)^2. > > They may also have other representations, but these suffice to justify > the statement.
How about justifying a slight variant of the above statement:
Statement: Every integer N that is 2 (mod 4) cannot be expressed as the difference of two squares.
One has the identity: a^2 - b^2 = (a+b)(a-b) .
That means the difference N of the squares of two integral numbers a and b can always be written as a product of the form (a+b)(a-b); a,b in Z:
N := a^2 - b^2 = (a+b)(a-b) ; a,b \in Z .
If one has a factorization N = Factor_1 * Factor_2 ; Factor_1, Factor_2 \in Z, then
Factor_1 = (a+b); Factor_2 = (a-b);
leads to a = (Factor_1 + Factor_2)/2 and b = (Factor_1 - Factor_2)/2.
Here a and b represent integral numbers only if Factor_1 and Factor_2 are of the same parity. That circumstance in turn implies that in that case N is constitutable as a product of two factors of equal parity which is not possible if N is an odd multiple of 2 / if N is 2 (mod 4). [N = 2 (mod 4) -> N = 4k + 2 = 2*(2k+1) , 2k+1 is odd.]
If (k-j) is even, then this expression represents an odd number and thus a number which is not 2 (mod 4) and where multiplying by -1 does not yield a number which is 2 (mod 4).
If (k-j) is odd, then (k-j+1) is even, then (k-j+1)*2j is 0 (mod 4) and (k-j+1)^2 is 0 (mod 4), thus in this case the entire expression represents a number which is 0 (mod 4) / a number which is not 2 (mod 4) and where multiplying by -1 does not yield a number which is 2 (mod 4).
In all the cases above, which are all cases, neither the absolute value of the difference of two arbitrary different squares, nor the product of that difference and (-1) is 2 (mod 4).