24Is it old ???
I dont know that..
just a try..I am not good at theory of numbers though..
I think there should be a condition taht n>1 becoz for n=1
it becomes 5 which is prime
anyways
if n is even
then 4n=even
and n4 is obviously even..so 4n+n4 is even
also it will be larger than 2....so it is a composite no.
Now If n is odd
4n+n4
if we put n=2m+1 in 4n
we get
42m4+n4
=>4.(2m)4+n4
now putting 2m=l
=>4l4+n4
=> (n2+2l2)2-(2nl)2
which is composite
Plz tell me of mistakes which I have committed...
1@eureka 1 doubt
when n is odd we can surly express dat as 2m+1
so we get 4n+n4=42m+1+(2m+1)4
which is equal to
( 24m+2+2m(some constant)+1)......on binomially expandin (2m+1)4
now temme how did u proove this to be composite!!
1When n is even and n>=2 then, the expression has a divisor 2.
When n is odd let n= 2k+1
n4+4n
=(2k+1)4+42k+1
=(2k+1)4+4.(2k)4
={(2k+1)2+2.22k}2 - 4{(2k+1)2k}2
={(2k+1)2+2.22k+2.2k(2k+1)}{(2k+1)2+2.22k-2.2k(2k+1)}
Hence the expression is composite.
Am i right ? Any shortcut method ?
24
eureka123
·2009-09-14 12:28:41
@ archna..I did that way becoz only 3 days back I read about Sophie Germain's Identity [3]
so I tried to bring the question to that form.......I dont think I commited any mistake in it...but still....i need experts approval..
62do you need to take the cases even and odd???
This is much more simple? or is it?
62
Lokesh Verma
·2009-09-15 01:30:16
Hint: Can you express the above as the difference of 2 squares?
11We know that for all nos N except of the form 4k+2 there exists integers x and y such that x^2-y^2=N
For some number representable as the difference of 2 squares - it can never represent a prime, as (x+y) or (x-y) or both shall divide it, meaning the above number cannot be a prime!
62
Lokesh Verma
·2009-09-15 02:26:43
soumik but that does not prove that N is prime..
what if x-y is 1 and x+y is prime
like 42-32=7?
34n+[(n+1)(n-1)]2+1+2(n+1)(n-1)
which is odd clearly but dunno how to prove that the above no is composite.can u give me hints to prove from this step sir.
we can easily prove by binomial expansion.
24
eureka123
·2009-09-15 07:03:23
but u agree naa that n >1
11
Devil
·2009-09-15 10:22:39
How on earth can I miss this?????
(n4+4n)=(n2+2n)2-2n+1n2
2n+1n2=(2a)2n2
I guess this finishes it!
