Landau's 4th problem asks if $\,L\_n=n^2 + 1\,$ is prime for infinitely many $n \in \Bbb{N}$. This obviously occurs only when $n$ is even.


Let's consider composite positive integers represented by
$$\,(n+1)^4+4=(n^2+1)\cdot((n+2)^2+1)=L\_n\cdot L\_{n+2}$$


If the number of Landau's primes was finite, let it be $L\_M$ the last one, then composite positive integers $\,k^4+4\,$ ($k$ odd) should admit at least 4 distinct prime factors for each $\,k\ge M+3\,$ (notice that $\,gcd(L\_n\,,L\_{n+2})=1$).


**Would it be possible to prove that the above statement is false, showing that at least a value of $\,k\,$ must exist for which the distinct prime factors of $\,k^4+4\,$ are less than 4, regardless the value of M?**


For example $\,125^4+4=244140629=15377 \cdot 15877$.


Many thanks.



About Landau's 4th problem
Suppose $k^4+4=L\_{k-1}L\_{k+1}$ has fewer than four prime divisors. Since the factors are relatively prime, this implies that one of them is a prime power. But since $L\_{k\pm 1}$ is one larger than a square, from Mihailescu's theorem (more specifically its special case when one of the powers is a square) we get that that this factor must in fact be prime. Therefore the assertion that $k^4+4$ infinitely often has at most $3$ prime factors implies Landau's 4th problem.


However (contrary to what I said in a comment), as far as I can tell, it is not equivalent to it. That said, let me present an argument that it *should* still be true. In fact, the conjecture Sylvain JULIEN dares to suggest in the comment should be true, that for any $n\geq 2$ we should have $\omega(k^n+n)<n$ infinitely often. This follows from [Schinzel's Hypothesis H](https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H). Indeed, let $d$ be the number of irreducible factors of $x^n+n$ (which is less than $n$, since there are no linear factors, and usually is $1$, e.g. if $n$ has some prime factor of exponent one). Since $x^n+n$ has no fixed prime factors (because for $x=0$ and $x=1$ the values are relatively prime), the Hypothesis H implies those factors should be simultaneously prime infinitely often, and thus $\omega(k^n+n)=d<n$ would hold infinitely often.


