math.NT — Number Theory

Nested Squares and Distinct-Prime Goldbach Partitions

Contributed by Frank Vega

The distinct-prime Goldbach conjecture asserts that for every $N\geq 4$ there is $M\in A_N=[1,N-3]$ with $N-M$ and $N+M$ both prime; geometrically, nested squares of side lengths $N$ and $M$ have an L-shaped difference of semiprime area $(N-M)(N+M)$. With $C_N$ and $D_N$ the sets of $M\in A_N$ for which $N-M$, respectively $N+M$, is prime, the conjecture for $2N$ is $C_N\cap D_N\neq\varnothing$. We embed this centered condition in a straddling one: $C_N\cap D_N$ lies in the set $E_N$ of half-differences $(Q-P)/2\in A_N$ of primes $2<P<N<Q<2N$. From Mikawa's theorem on prime pairs with almost all differences we deduce $|A_N\setminus E_N|\ll_D N/(\log N)^D$ for every $D>0$, and with explicit constants we prove $|E_N|>N/760$ for every $N\geq 3275$. Consequently, for all large $N$ and all but $O_D(N/(\log N)^D)$ primes $P<N$, the number $2N-P$ becomes prime after adding the difference $P'-P$ of two primes $P'<N$; Goldbach's partition is the case $P'=P$. In contrast, $|C_N\cap D_N|<(N-3)-\log^2(2N)$ for every $N\geq 34$, and covering all but $\log^2(2N)$ values of $A_N$ for all large $N$ would imply the Cram\'er-type bound $p'-p<2\log^2(2p+2)+2$. The method also shows that $2N$ is a sum of two distinct primes for more than $X/9600$ integers $N\leq X$, for every $X\geq 10^6$. Computations up to $N=10^6$ compare the number of distinct-prime partitions with the Hardy--Littlewood prediction and measure the missing values of $E_N$ forced by prime gaps. The conjecture itself is not proved here.