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$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times 3\times23 \\ 1105 & = 5\times13\times17 \\ 1106 & = 2\times7\times79 \end{align} $$ In looking at this list of prime factorizations, I see that all of the first 10 prime numbers, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, appear within the factorizations of only seven consecutive integers. (The next prime number, 31, has its multiples as far from 1100 as it could hope to get them (1085 and 1116).) So no nearby number could hope to be divisible by 29 or 23, nor even by 7 for suitably adjusted values of "nearby". Consequently when you're factoring nearby numbers, you're deprived of those small primes as potential factors by which they might be divisible. So nearby numbers, for lack of small primes that could divide them, must be divisible by large primes. And accordingly, not far away, we find $1099=7\times157$ (157 doesn't show up so often---only once every 157 steps---that you'd usually expect to find it so close by) and likewise 1098 is divisible by 61, 1108 by 277, 1096 by 137, 1095 by 73, 1094 by 547, etc.; and 1097 and 1109 are themselves prime.

So if an unusually large number of small primes occur unusually close together as factors, then an unusually large number of large primes must also be in the neighborhood.

Are there known precise results quantifying this phenomenon?

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    Here is a way to rephrase your question: Let $P(n)$ be the largest prime factor dividing $n$. Then we are looking at the statistics of $P(n)$. In particular, your statement corresponds to a statement concerning the average of $P(n)$ on a short interval. If we can prove that in a particular neighborhood of $x$, $P(n)$ will always have a particular average, then it follows that if I have a sequence of $n$ such that $P(n)$ is small, then on some larger interval I will have a disproportionate number of $n$ such that it is large.2011-10-18
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    Hmmm, well, it's been proven (by both Erdos and Chebyschev, I believe) that for $n\in \mathbb{N}$, there's always a prime between $n$ and $2n$. So, if your intervals get large enough, then you'll have a fixed point of $P$ (ie a prime argument) in the interval, which will be a relatively large value of $P$. I'm not sure that helps a lot, but it was the first observation off the top of my head.2011-10-18
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    Ok, I am pretty sure that using theorems related to $\Psi(x,y)$ we can calculate the average of $P(n)$ around $x$ in the interval $\left(x,x+x\exp(-c\sqrt{\log x})\right)$. Maybe there is a way to do better, but I am doubtful. (Of course you get $x^{{1}{2}+\epsilon}$ on RH). I believe that the average you get is asymptotic to $\frac{\pi^2}{12} \frac{x}{\log x}$.2011-10-18
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    You're asking us to do a lot of work. To make any sense of your question, we have to quantify what you mean by small primes, and what you mean by large primes, and how close is unusually close, and how many small primes it takes to be an unusually large number, and what's a neighborhood, and how many large primes it takes to be an unusually large number. And then we still have to answer the question! So it's only fair to ask: is it worth it? Is there any payoff? Will you be able to use the answer to do something else?2011-10-19
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    @Eric: Could you remind me? What is $\Psi(x,y)$?2011-10-19
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    @Gerry: My question was "Are there known precise results quantifying this phenomenon?" To know the answer to that question is to know that all that hard work has already been done. But I _don't_ know that; hence my question.2011-10-19
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    Just to make it doubly clear to whom it may concern that you were not asking for extra work, I have added the "reference-request" tag to your excellent question.2011-10-20
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    Question also posted to MathOverflow, http://mathoverflow.net/questions/78626/small-primes-attract-large-primes2011-10-20
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    I have a perfect answer to your question, your question reminds me of a paper that I read some months before, a paper by K.Ramachandra who was a seminal mathematician of India, in that paper we can find some part of your question that when a number is factorized into prime factors then what is the distance between those prime factors, and also there has been many theorems proved in such case , I hope I surely post the answer in some days as soon as I get the name of paper I saw. @MichaelHardy2012-01-11

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