Try this & check ur strength?

Question

Prove that any number having any power can be expressed as difference of two squares & also prove that the numbers of the form 4t+2 canot be expressed as difference of two squares.

like 3^(3)= 14^(2)-13^(2)

Answer 1

both are easy

any number having any power > 1(for power 1 for the form 4t+2) is not possible

it is odd

say 2n+1 = (n+1)^2 - n^2

it is even then it is divisible by 4^n . if is not divisible by 4^n then it is 4^n. 2 and mth root cannot be taken.
let highest power of 2 is 2^2n because it cannot be odd

so 2^n.odd

odd can be subtract of 2 perferct squares.

so multiply both by 2^n so even powe can be perfect of 2 squares

b)
the number is of the form 4n+2

difference of 2 squares say a and b

a can be 2n or (2n+1) =>a^2 = 4n^2 or 4n^2+4n+1
b can be 2m or 2m+1 =>b^2 = 4m^2 or 4m^2+4m+2


a^2 mod 4 = 0 or 1
b^2 mod 4 = 0 or 1
a^2 -b^2 mod 4 = 0-0 or 1-0 or 0-1 or 1-1
= 0 or 1 or -1 or 0

so it cannot be 2
so it is not of the form 4t+2

Answer 2

For second question:
4t + 2 = ?
Let t = -0∙5
4t + 2 = ?
4(-0∙5) + 2 =
-2 + 2 = 0
Can zero be expressed as the difference of two squares ?

Answer 3

is the ansewr a number or a number with a power or what. That confused me. My streanght is apperantly a zero.

Answer 4

I expected a question on keep fit!.........pass!

Answer 5

No, I wont help with your homework ;-)