Given a prime $p$, let $P$ be the product of the primes less than $p$. A $p$-simple integer $s$ is then an integer satisfying $\gcd(P,s)=1$, and the posted conjecture asserts that any interval of $p$ consecutive integers contains at least one such $s$, and this means the difference between two consecutive totatives of $P$ is at most $p$.

This holds for $p\leq 11$ (as $g(210)=10$), but not for larger $p$, as the example in the comments with $p=13$ and $n=114$ shows. The Jacobsthal function $g(m)$ which measures the maximal difference between consecutive totatives of $m$ is larger than $k=\Omega(m)$, the number of distinct prime factors of $m$, and when restricted to primorials $P$ is easily shown to grow at least as fast as $2q$, where $q$ is the largest but one prime less than $p$. As mentioned in another answer, Rankin (and earlier, Westzynthius in 1931) showed that for any positive $C$, there are some $m$ with $g(m) \gt Ck\log k$.

Gerhard "Now I've Mentioned Jacobsthal's Function" Paseman, 2016.06.16.

Given a prime $p$, let $P$ be the product of the primes less than $p$. A $p$-simple integer $s$ is then an integer satisfying $\gcd(P,s)=1$ (so $s$ is a totative of $P$), and the posted conjecture asserts that any interval of $p$ consecutive integers contains at least one such $s$, and this means the difference between two consecutive totatives of $P$ is at most $p$.

This holds for $p\leq 11$ (as $g(210)=10$), but not for larger $p$, as the example in the comments with $p=13$ and $n=114$ shows. The Jacobsthal function $g(m)$ which measures the maximal difference between consecutive totatives of $m$ is larger than $k=\Omega(m)$, the number of distinct prime factors of $m$, and when restricted to primorials $P$ is easily shown to grow at least as fast as $2q$, where $q$ is the largest but one prime less than $p$. As mentioned in another answer, Rankin (and earlier, Westzynthius in 1931) showed that for any positive $C$, there are some $m$ with $g(m) \gt Ck\log k$.

Gerhard "Now I've Mentioned Jacobsthal's Function" Paseman, 2016.06.16.

Given a prime $p$, let $P$ be the product of the primes less than $p$. A $p$-simple integer $s$ is then an integer satisfying $\gcd(P,s)=1$, and the posted conjecture asserts that any interval of $p$ consecutive integers contains at least one such $s$, and this means the difference between two consecutive totatives of $P$ is at most $p$.

This holds for $p\leq 11$ (as $g(210)=10$), but not for larger $p$, as the example in the comments with $n=114$ shows. The Jacobsthal function $g(m)$ which measures the maximal difference between consecutive totatives of $m$ is larger than $k=\Omega(m)$, the number of distinct prime factors of $m$, and when restricted to primorials $P$ is easily shown to grow at least as fast as $2q$, where $q$ is the largest but one prime less than $p$. As mentioned in another answer, Rankin (and earlier, Westzynthius in 1931) showed that for any positive $C$, there are some $m$ with $g(m) \gt Ck\log k$.

Gerhard "Now I've Mentioned Jacobsthal's Function" Paseman, 2016.06.16.

Given a prime $p$, let $P$ be the product of the primes less than $p$. A $p$-simple integer $s$ is then an integer satisfying $\gcd(P,s)=1$, and the posted conjecture asserts that any interval of $p$ consecutive integers contains at least one such $s$, and this means the difference between two consecutive totatives of $P$ is at most $p$.

This holds for $p\leq 11$ (as $g(210)=10$), but not for larger $p$, as the example in the comments with $p=13$ and $n=114$ shows. The Jacobsthal function $g(m)$ which measures the maximal difference between consecutive totatives of $m$ is larger than $k=\Omega(m)$, the number of distinct prime factors of $m$, and when restricted to primorials $P$ is easily shown to grow at least as fast as $2q$, where $q$ is the largest but one prime less than $p$. As mentioned in another answer, Rankin (and earlier, Westzynthius in 1931) showed that for any positive $C$, there are some $m$ with $g(m) \gt Ck\log k$.

Gerhard "Now I've Mentioned Jacobsthal's Function" Paseman, 2016.06.16.