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There is no known explicit relation between the prime factors of $n+1$ given the prime factorization of $n$.

In fact, this is considered one of the hardest problems in our current understanding of number theory. Paul Erdős once famously quoted, albeit within the context of the Collatz conjecture - closely linked to the prime factorization of consecutive integers - that:

"Mathematics is not yet ready for such problems".

We can however deduce some basic properties for $n+1$ in the following way. If we denote by $\omega(n)$ the number of distinct prime factors of $n$ and by $\alpha_i$ the multiplicity of the $i^{th}$ prime in its decomposition, we have:

$$ n = \prod_{i=1}^{\omega(n)} p_i^{\alpha_i}. $$

From this we get:

$n+1$ is not divisible by any of the $p_i$ , otherwise $p_i$ would divide $(n+1)-n$ , which is clearly impossible because no prime number divides $1$ ,

is not divisible by any of the , otherwise would divide , which is clearly impossible because no prime number divides , If $n$ is even, $n+1$ is odd and vice-versa, which can trivially be extended to congruence modulo $p_i$ ,

is even, is odd and vice-versa, which can trivially be extended to congruence modulo , There is no obvious relation between $\omega(n)$ and $\omega(n+1)$ .

One less obvious "observation" is Wilson's theorem. It states that if $p$ is a prime number, we have the following congruence relation:

$$ (p-1)!\ \equiv\ -1 \pmod p $$

This connects the prime $p$ with its immediate integer predecessor $p-1$.

There are also some non-trivial observations made by Erdős and Pomerance$^1$, which are the following. Define $P(n)$ as the largest prime factor of $n$. Then:

$P(n)>P(n+1), P(n+1)>P(n+2), P(n)<P(n+1)$ and $P(n+1)<P(n+2)$ occur infinitely often,

and occur infinitely often, $P(n)>P(n+1)>P(n+2)$ does not occur infinitely often,

does not occur infinitely often, $P(n)$ and $P(n+1)$ are usually not close, i.e., for each $\epsilon>0$ , there is a $\delta>0$ such that for sufficiently large $x$ , the number of $n\leq x$ with

$$ x^{-\delta}<P(n)/P(n+1)<x^{\delta} $$

is less than $\epsilon x$.

Any integer $n\leq x$ is divisible by at most $\log(x)/\log(2)$ primes.

$^1$ Erdős and Pomerance, "On the largest prime factors of $n$ and $n+1$", Aequationes Mathematicae 17 (1978), available at: https://www.math.dartmouth.edu/~carlp/PDF/paper17.pdf