Math
Integer interval $[p_n,p_{n+1}]$ has a prime factor $y_n$ such that $y_n > p_{n+1} - p_n - 1$?
Step-by-step mathematics solution: Integer interval $[p_n,p_{n+1}]$ has a prime factor $y_n$ such that $y_n > p_{n+1} - p_n - 1$?
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1. What the problem is asking (in plain language)
Let
[ p_n<p_{n+1} ]
be two consecutive odd primes and put
[ g_n = p_{n+1}-p_n \qquad(\text{the prime gap}). ]
The interval
[ [p_n ,\,p_{n+1}] ]
contains the two primes themselves and the (g_n-1) composite numbers that lie between them.
Claim T (the statement we have to check) says:
In every such interval there is a composite integer (x_n) whose largest prime divisor
[ y_n = P^{+}(x_n) ] satisfies
[ \boxed{ \; y_n > g_n-1 \;} ]
In other words: no matter how large the gap between two consecutive odd primes is, one of the numbers that sits inside the gap must have a prime factor that is larger than the length of the gap (minus 1).
The question is whether this statement is a known theorem, a conjecture, or already disproved by a counter‑example.
2. Reformulation in terms of “smooth” numbers
A positive integer is called (y)-smooth if all of its prime factors are (\le y).
Denote by
[ P^{+}(m)=\max{ \text{prime divisor of } m} ]
the largest prime divisor of (m).
Then the condition (P^{+}(x_n) > g_n-1) is exactly the negation of
[ x_n\;\text{is }(g_n-1)\text{-smooth}. ]
Hence Claim T is equivalent to the following statement:
Smooth‑gap conjecture.
For every pair of consecutive odd primes (p<p’) the set
({p,p+1,\dots ,p’}) contains at least one integer that is not ((p’-p-1))-smooth.
If the conjecture were false, we would have a prime gap (g) together with a run of (g-1) consecutive integers, all of whose prime factors are at most (g-1). Such a run is called a smooth run of length (g-1).
3. What is known about long smooth runs?
-
Existence of arbitrarily long runs of composites.
Classical constructions (e.g. (n!+2, n!+3,\dots , n!+n)) show that for every (k) there are (k) consecutive composite numbers.
However the numbers (n!+m) are not guaranteed to be (k)-smooth; they may have a prime divisor larger than (k). -
Upper bounds for smooth runs.
Let ( \Psi(x,y) ) denote the number of (y)-smooth integers (\le x). A deep result of de Bruijn (and later refinements by Hildebrand, Tenenbaum, etc.) gives[ \Psi(x,y)=x\,\rho!\bigl(\frac{\log x
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