certain a^(n)-b^(n) can be written as c^(2)-d^(2) like 7^(3)-4^(3) = 48^(2)-45^(2) (the numbers of the form of 4t+2 are not expressible as difference of two squares like 26=3^(3) - 1^(3) ). In such cases can we write the power as 2 in place of "n"?
2006-10-27 21:55:46 · 8 answers · asked by rajesh bhowmick 2 in Science & Mathematics ➔ Mathematics
Well, the expression 4t + 2 tells us that IT IS an even integer which is NOT divisible by 4. Let me show you this:
--------------------------------------------
Let
a = a first integer, and
a + d = a second integer.
Their difference is d.
Thus, the difference of their squares is:
(a + d)² - a²
= a² + 2ad + d² - a²
= 2ad + d²
Now, we only have 2 choices for d: it is either (a) d is odd, or (b) d is even. Let us consider both cases:
________________________
(a) Now, if d is odd, note that we can rewrite d = 2k + 1 to be an odd integer, and then the difference of the squares is:
= 2a(2k + 1) + (2k + 1)²
= 4ak + 2a + 4k² + 4k + 1
= 2(2ak + a + 2k² + 2k) + 1
= 2j + 1 = odd integer
________________________
(b) Now, if d is even, note that we can rewrite d = 2k to be an even integer, and then the difference of the squares is:
= 2a(2k) + (2k)²
= 4ak + 4k²
= 4(ak + k²)
= 4j = even integer divisible by 4.
________________________
Thus, we can infer that the difference of 2 squares is always either an odd integer, or an even integer divisible by 4.
Thus, it CANNOT be an even integer NOT divisible by 4, as what 4t + 2 implies.
The expression aⁿ - bⁿ is always an integer, so as long as it is not an even integer NOT divisible by 4, it can be expressed as a difference of 2 squares.
Hope you understand... it is true.
^_^
^_^
2006-10-27 23:36:58 · answer #1 · answered by kevin! 5 · 1⤊ 0⤋
2
2006-10-28 05:09:42 · answer #2 · answered by Anonymous · 0⤊ 0⤋
then
a^(2)-b^(2) = c^(2)-d^(2)
or, a^(2) + d^(2) = b^(2) + c^(2)
now are two right triangles equal if only their hypotenuse are equal? Nope, we get infinitely many right angle triangles with equal hypotenuse.
So n can be easily replaced by 2
2006-10-28 05:08:04 · answer #3 · answered by The Potter Boy 3 · 0⤊ 0⤋
since
'a' is not equal 'c'
'b' is not equal 'd' or(vise versa)
we can use 2 as power in the place of 'n'
by
waste fellow
2006-10-28 05:31:03 · answer #4 · answered by waste fello 1 · 0⤊ 0⤋
yes
2006-10-28 05:14:14 · answer #5 · answered by gjmb1960 7 · 0⤊ 0⤋
the math dude again :P !!
2006-10-28 04:57:31 · answer #6 · answered by dodi 3 · 0⤊ 0⤋
hadache
2006-10-28 05:26:01 · answer #7 · answered by Devil May Cry 6 · 0⤊ 1⤋
Sure, why not?
2006-10-28 05:09:03 · answer #8 · answered by sueno 1 · 0⤊ 0⤋