It is known that $a_1, a_2,$...$, a_{50}$ is an arithmetic sequence with common difference $d$, and $a_i (i=1, 2, ..., 50)$ are primes. If $a_1>50$, prove that $d>600,000,000,000,000,000.$
Honestly I have completely no idea how to solve this question. I don't even know how $6*10^{17}$ is derived from. Does anyone know how to solve this?
Fix any prime $p<50$, and consider the sequence mod $p$. Notice that none of the terms (say $a$) can be equal to $0$, because otherwise $p\mid a$. But if $p\nmid d$, the sequence will eventually be $0$ mod $p$. (within the first 50 terms). Hence $p\mid d$ for all primes less than 50.
If you multiply all the primes out, you get around $6.15\times10^{17}$ which is large enough.