If $n > 1$ and all $n$ positive integers $a, a + k, \cdots , a+ (n - 1)k$ are odd primes, show every prime $Background: This is from Rosen 5th edition, $3.2.15$ Number Theory.
This is an important proof because $3$ following problems require it to be correct.

If $i=0$, and $j=p$ then this proof is wrong and $p\mid (i-j)$ and $p$ may not divide $k$. Can this proof be modified so that it is correct or do I not understand it?

If $i=0$, then $p$ would have to be equal to $-j$. This would be a contradiction since $i, j, p > 0$. WLOG, we can assume $i-j>0$.
Moreover, since $i \le p$ and $j \le p$, then the claim that $i-j < p$ holds, and the proof is true. Does that help?
