# Distribution of Goldbach's weak-conjecture's prime-triples

From Harald Helfgott's proof of Goldbach's weak conjecture,1 we know that every odd number $$> 7$$ is the sum of three odd primes. If $$n$$ is such an odd number, say that two sums that yield $$n$$ are distinct if they are distinct as sorted lists. So $$n=13$$ has just two distinct sums, $$(3,3,7)$$ and $$(3,5,5)$$.

Q1. Are $$9=3+3+3$$ and $$11 = 3+3+5$$ the only two odd numbers $$> 7$$ that have unique sums of three odd primes?

Call a sum of three odd primes as a $$3$$-sum for conciseness. Here are the first few odd numbers and their $$3$$-sum lists: $$\left| \begin{array}{ccc} n & 3{-}sums & \#3{-}sums\\ \hline 9 & (3,3,3) & 1\\ 11 & (3,3,5) & 1\\ 13 & (3, 3, 7),\; (3, 5, 5) & 2\\ 15 & (3, 5, 7),\; (5, 5, 5) & 2\\ 17 & (3, 3, 11),\; (3, 7, 7),\; (5, 5, 7) & 3 \end{array} \right|$$

Here is a plot of the number of $$3$$-sums for odds up to about $$1001$$:

Q2. What explains the separation into two distinguishable series?

So for $$n=975$$, there are $$701$$ $$3$$-sums:

$$\begin{eqnarray} &(3, 5, 967)\\ &(3, 19, 953)\\ &(3, 31, 941)\\ &\cdots\\ &(313, 313, 349)\\ &(313, 331, 331) \end{eqnarray}$$

But for $$n=977$$, there are $$1106$$ $$3$$-sums. Here is a bit more data: $$\left| \begin{array}{cc} n & \#3{-}sums \\ \hline 975 & 701 \\ 977 & 1106 \\ 979 & 1010 \\ 981 & 758 \\ 983 & 1113 \\ 985 & 977 \\ 987 & 756 \\ 989 & 1110 \\ 991 & 1037 \\ 993 & 791 \\ 995 & 1056 \\ 997 & 1096 \\ 999 & 770 \\ 1001 & 1095 \\ \end{array} \right|$$

1MO: Proof of the weak Goldbach Conjecture .

• Did you think about mod 3 considerations? Gerhard "Can You Finish The Thought?" Paseman, 2019.06.18. – Gerhard Paseman Jun 19 '19 at 0:06
• Presumably Helfgott and Platt's computations determined that the answer to Q1 is "yes" among the first 10-to-the-umpteen integers, and Helfgott's explicit bounds for larger integers show that the answer to Q1 is "yes" for the rest. – Greg Martin Jun 19 '19 at 0:32
• Have you considered the integers 'many' 3-sums of which are of the form $(p, p, q)$ with $p\neq q$? – Sylvain JULIEN Jun 19 '19 at 10:09
• Are you familiar, Joseph, with "Goldbach's Comet"? – Gerry Myerson Jun 19 '19 at 13:05
• @GerryMyerson: I did not know about Goldbach's comet. What a great name! Clearly my plot is a (monochromatic) version of the comet for triples rather than pairs. I appreciate learning of this---Thanks! – Joseph O'Rourke Jun 19 '19 at 14:49