Composites Between Adjacent Primes

[franktaw at netscape.net]

The answer is almost certainly yes.  See "good" sequences (A090318).  While an exception would not necessarily have to be a good sequence, it would have to be very close to being one (it could have the largest prime divisor repeated near the beginning and end, with a slightly smaller one alone in the middle).  And the good sequences appear to grow fast enough that there are always going to be larger primes dividing some members of the sequence.
 
This might be provable using analytical methods.  Not by me, though.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: Leroy Quet qq-quet at mindspring.com

...
Is there always only one composite between each pair of adjacent primes, 
p(n) and p(n+1), which is divisible by the highest prime dividing any 
composite between p(n) and p(n+1)?
...
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