Monday, May 10, 2010

Finite Prime Set

Let $P$ be a finite set of primes. Define $m(P)$ as the largest number of consecutive integers each of which is divisible by a prime in $P$.
Let $|P|$ denote the size of $P$ and $\min(P)$ to be the smallest element in $P$.

Prove that:
1. $m(P) \geq |P|$
2. $m(P) = |P|$ if and only if $\min (P) > |P|$


Let $s = |P|$. We proceed by constructing $s$ consecutive integers which are each divisible by a prime in $P$. Indeed, the following system has a solution, according to Chinese Remainder Theorem:
$x+1 \equiv 0 \pmod {p_1}$
$x+2 \equiv 0 \pmod {p_2}$
$x+s \equiv 0 \pmod {p_s}$

Second part:
Suppose we have $s < \min(P)$ and that we have $s+1$ consecutive integers all divisible by some $p_i$: $x, x+1, x+2, ... , x+s$. Let $p_1 = \min(P)$.

For each $p_i$, it can only divide at most one of the above-mentioned integers. For if it divides $x+a$ and $x+b$ then it also divides $|a-b| \leq s < p_1$, a contradiction (since $p_1$ is the smallest prime in $P$).
So $m(P) \leq s$. But it's been shown that $m(P) \geq s$, so $m(P) = s$.

Now suppose we have $s \geq \min(P)$. We will show $s+1$ consecutive integers such that each is divisible by a prime in $P$, which then establishes $m(P) > s$.
Let $k = \min(P)$.
Again, we set up a system of equations as described above, but we choose the ordering of $p_i$s such that $p_k = k$. The rest can be arbitrary ordering. This is always possible if $s \geq k$.
According to CRT, there is a solution of $s$ consecutive integers that satisfy the above system of equations. But then we also have $x = x+k - k$ divisible by $p_k = k$. So together with $x$, they form $s+1$ consecutive integers.

No comments:

Post a Comment