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For a positive integer $x$ let us write $A_x$ to denote the set of all the prime factors of $x$.

Conjecture. For any given positive integer $n$ there is $N$ large enough such that whenever $x$ is a positive integer with $x\geq N$ we have $A_x\neq A_{x+n}$.

Suppose $x$ is such that $A_{x}=A_{x+n} = \set{p_1,\ldots, p_k}$. Then each $p_i$ must divide $n$. Write $$ x=p_1^{\alpha_1} \cdots p_k^{\alpha_k},\quad x+n= p_1^{\beta_1} \cdots p_k^{\beta_k} $$ Then the equation $(x+n) - x= n$ gives $$ p_1^{\beta_1} \cdots p_k^{\beta_k} - p_1^{\alpha_1} \cdots p_k^{\alpha_k} = n $$ This seems to suggest the following conjecture. Let $S=\set{p_1,\ldots, p_k}$ be a set of primes. For $\alpha = (\alpha_1,\ldots, \alpha_k)\in \N^k$, we write $p^\alpha$ to mean the number $p_1^{\alpha_1} \cdots p_k^{\alpha_k}$. Then

Conjecture. For any given natural number $n$, there are only finitely many distinct elements $\alpha, \beta\in \N^k$ such that $|p^\alpha-p^\beta|\leq n$.

(Cancelling out from LHS and RHS in the equation $p^\alpha-q^\beta = \pm n$, we may assume that $\sum_i \alpha_i\beta_i=0$.)

In particular, the conjecture above, if true, implies that for any given $n$, the gap $|2^\alpha-3^\beta|$ is less than $n$ for only finitely many pairs $(\alpha, \beta)\in \N^2$, which seems intuitively reasonable but I am not able to make any progress.

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  • $\begingroup$ oeis.org/A219794 gives "First differences of 5-smooth numbers". It says, "Thue-Siegel-Roth theorem gives $\lim a(n) = \infty$" which says the difference between numbers divisible only by primes $2,3,5$ goes to infinity. "With the aid of lower bounds for linear forms in logarithms, Tijdeman showed that $a(n+1)-a(n) > a(n)/(\log a(n))^C$ for some effectively computable constant $C$. The reference is R. Tijdeman, On integers with many small prime factors, Compos. Math. 26 (1973), 319--330. What goes for $5$-smooth surely goes for $B$-smooth. $\endgroup$ Aug 1, 2020 at 12:30

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R. Tijdeman, On integers with many small prime factors, Compositio Mathematica 26 (1973) 319-330, says:

Let $p$ be a prime, $p\ge3$, and let $n_1=1<n_2<\cdots$ be the sequence of all positive integers composed of primes $\le p$. [...] Thue derived from his famous result on the approximation of algebraic numbers by rationals that $\lim_{i\to\infty}(n_{i+1}-n_i)=\infty$.

This is already enough to prove the "Conjecture". Tijdeman goes on to improve on Thue's result. The Thue reference is Bermerkungen uber gewisse Naherungsbruche algebraischer Zahlen, Skrift. Vidensk. Selsk. Christ., 1908, Nr. 3.

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  • $\begingroup$ Thank you. Very helpful. I will accept in some time. $\endgroup$ Aug 1, 2020 at 13:10

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