This is related to the conjecture that all odd integers greater than $17$ can be written as the sum of 3 distinct primes.

Schinzel showed that the Goldbach conjecture implied this in 1959 and as the Goldbach conjecture has been verified up to $4\times10^{18}$ by Oliveria e Silva, Herzog and Pardi, this conjecture holds up to there as well.

Vinogradov's proof that all sufficiently large odd integers are the sum of three primes implies that as the number of representations of a sufficiently large odd integer as the sum of three primes is large enough, it must be the sum of 3 distinct primes.

Harald Helfgott's proof of the weak Goldbach conjecture also implies that all odd integers greater than $10^{27}$ can be represented as the sum of 3 distinct primes (assuming I've understood it correctly).

So the interval between $4\times 10^{18}$ and $10^{27}$ is the remaining interval on which to verify whether all odd integers greater than $17$ can be written as the sum of 3 distinct primes.

In Helfgott's proof, he uses the fact that the Goldbach conjecture is verified up to $4\times 10^{18}$ along with a prime ladder to show that the odd integers in this interval are the sum of 3 primes. But that prime ladder is a list of primes from 3 to beyond $10^{27}$ such that consecutive primes have difference at least 6 and at most $4\times 10^{18}$. Thus for odd $n$ between $4\times 10^{18}$ and $10^{27}$, there is always a $p$ in the ladder such that $n-p$ is equal to an even number less than $4\times 10^{18}$ and is therefore the sum of 2 primes, $q$ and $r$, so $n = p + q + r$.

As the verification of the Goldbach conjecture up to $4\times 10^{18}$ showed that each even integer was the sum of 2 distinct primes, it would be necessary to create a similar prime ladder such that consecutive primes had difference at least $8$ (as $n-p=6$ could only be written as 3 primes as $n = p + 3 + 3$ which is not a representation with distinct primes) and ensuring that odd integers between $4\times 10^{18}$ and $8\times 10^{18}$ are the sum of 3 distinct primes (as $p$ could potentially equal $q$ or $r$ for $n$ in this interval).

How difficult is it to make such a ladder and verify that primes between $4\times 10^{18}$ and $8\times 10^{18}$ are the sum of three distinct primes? For the second part I imagine finding a prime, $s$, just below $4\times 10^{18} - 8$ and a prime, $t$, just above $4\times 10^{18}$ would solve it as by adding any even number less than $s$ to $s$ you could reach any odd number up to just below $8\times 10^{18}$ and by adding even numbers less than $t$ to $t$ you could reach the other odd numbers less than $8\times 10^{18}$.

3more comments