# Prime divisors of two large integers which are close.

$$\newcommand{\set}[1]{\{#1\}}$$ $$\newcommand{\N}{\mathbb N}$$

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.

• 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. Aug 1, 2020 at 12:30

Let $$p$$ be a prime, $$p\ge3$$, and let $$n_1=1 be the sequence of all positive integers composed of primes $$\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$$.