I am trying to prove:
Given any seventeen consecutive integers, there does not exist one which is coprime to the rest.
I am aware S.S.Pillai proved a similar statement for $m$ consecutive integers, where $17\leqslant m \leqslant 430$. I can't seem to find the proof online. Does anyone know where I can find the proof?
You seem to be garbling Pillai's result. Using the noation of http://www.math.tifr.res.in/~saradha/st-proc-r-f2.pdf (and indeed everything in my answer is taken from the first page of that paper), say a set $S$ has property $P_1$ if there is some $x\in S$ which is coprime to all other elements of $S$. Pillai proved two things:
Any set of consecutive integers with fewer than 17 elements has property $P_1$.
For $17\le m\le 430$, there are infinitely many sets $S$ of $m$-many consecutive integers where $S$ does not have property $P_1$. EDIT: The first example is the sequence of length $17$, starting with $2184$ (see https://oeis.org/A090318/internal).
Note that as fleablood says, what you've written is false for any $n>1$ at all: if $X$ is a set of $n$-many consecutive integers with largest element prime, then $p$ is coprime to all other elements of $X$.
(Note that it is not enough for $X$ to merely contain a prime - e.g. in $\{2, 3, 4, . . . , 18\}$, $2$ is prime but not coprime to the other elements!)
Note: Pillay later extended the second clause to $17\le m\le 12335$. Scott then extended this to $17\le m\le 2491906561$, and then finally Brauer got the full result: if $17\le m$, then there are infinitely many sets $S$ consisting of $m$-many consecutive integers, such that $S$ does not have property $P_1$. So the full result is:
If I have a set $S$ of $m$-many consecutive integers for $m<17$, then there is an element of $S$ which is coprime to the others.
However, I'm not guaranteed to be able to pull this off for $m\ge 17$; indeed, there will be infinitely many counterexamples (although of course there are also sets of arbitrarily many consecutive integers which do contain an element coprime to the others, namely any set containing at least one prime).
As for proofs, Brauer's paper (proving the full result) is publicly available at https://projecteuclid.org/download/pdf_1/euclid.bams/1183503578; Pillai's original papers "On $M$ consecutive integers I-IV" seem harder to find.