Are there any two identical terms in this series, defined parallely to the primes? Let $p_n$ denote $n$-th prime number and $k_n$ be sequence that is
\begin{align}
k_1 &= 1 \\
k_2 &= p_2 - k_1 &&( 3-1 = 2 ) \\
k_n &= p_n - k_{n-1} &&\text{( n is integer larger than 1 )}
\end{align}
this is result
$$1 , 2 , 3 , 4 , 7 , 6 , 11 , 8 , 15 , 14 , 17 , 20 , 21
, 22 , 25 , 28 , 31 , 30 , 37 , 34 , 39 , 40 , 43 , 46
, 51 , 50 , 53 , 54 , 55 , 58 , 69 , 62 , 75 , 64 , 85
, 66 , 91 , 72 , 95 , 78 , 101 , 80 , 111 , 82 , 115 ...$$
This is not a monotonically increasing sequence. My question is if there are two identical terms in this series. My guess is no.
Until $k_{200}$ still no special case appeared and until $k_{1000}$ and $k_{20000}$ too, 
I want to know that when is the first time the case appears?
I'm young kid and just curious but i belive that you smart people can solve this.
 A: Suppose $k_n = k_{n+1}$.
Then $k_n = p_{n+1} - k_n$ so $2k_n = p_{n+1}$.  Thus $p_{n+1}$ is an even prime.  $p_1 = 2$ is the only even prime and that can not happen as $n > 1$.  So this never happens.
Note $k_{n} = p_n - k_{n-1}$.  As $p_n$ is odd and "odd - odd = even" and "odd - even = odd".  The terms must switch from odd to even every turn.
Suppose $k_n = k_{n + i}$....
Let's look closer at what $k_n$ is.
$k_1 = 1$
$k_2 = p_2 - 1$
$k_3 = p_3 - p_2 + 1$
$k_4 = p_4 - p_3 + p_2 - 1$
....
$k_n = p_n - p_{n-1} + ....+p_2 -1 $  if $n$ is even.
$k_n = p_n - p_{n-1} + ....-p_2 +1 $  if $n$ is even.
So if $k_n = k_{n+i}$ because odd and even alternate $i$ is even and $k_n = k_{n+i}$ are either both even or both odd.
So $k_n = p_n - ..... \pm p_2 \mp 1$ and $k_{n+i}= p_{n+ i} - ..... \pm p_2 \mp 1$
$k_{n+i} = k_n \implies k_{n+i} - k_n = 0 \implies (p_{n+ i} - p_{n+i - 1}) + (p_{n+i - 2} - p_{n+i - 3}) + .... + (p_{n+2} - p_{n-1})=0$.
As $p_k > p_{k-1}$ each of the terms in parenthesis is positive.  So $k_{n+i} - k_n > 0$.  
So this sequence will never repeat.
A: Fleablood took care of the details of the proof. So, I'll just offer some exposition on his proof.
We split your series into two different series. One series consisting of odd terms $O$ , and one series consisting of even terms, $E$. We discover, (quite delightfully) that both $O$ and $E$ are monotonically increasing sequences, even if $K=\{k_n\}_0^\infty$ is not ! So, all terms in $E$ and $O$ are unique.
Now, it just remains to be shown that these two sequences never intersect. This is easily done.
