I have a question which belongs to the field of number theory. Can we prove or disprove the following claim:

For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one positive integer number $k$, such that the expression $\frac{k(n+1)}{7k-(n+1)}$ is a integer number.

I checked the above claim for prime numbers up to $17\times 10^9$, and the calculations verified it.

I will be so thankful for any helpful comments and answers.

positiveinteger. Now the claim is true, since one may apply the same scheme with taking $-4,-2,-1$ into account. But I wouldnt say that changing the question without any notification in such situation is a good way of doing things. $\endgroup$ – Ilya Bogdanov Dec 22 '15 at 16:22