Introduction
The Goldbach conjecture, proposed in 1742, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers [1 ] . Despite centuries of effort and computational verification up to 4 × 10 18 4 \times10^{18} 4 × 1 0 18 [2 ] , the conjecture remains unproven. This paper studies a natural variant: every even integer 2 N ≥ 8 2N \ge8 2 N ≥ 8 distinct primes. Our variant requires the two primes to be distinct, thus excluding the trivial representations 4 = 2 + 2 4 = 2 + 2 4 = 2 + 2 6 = 3 + 3 6 = 3 + 3 6 = 3 + 3 N ≥ 4 N \ge4 N ≥ 4 2 N ≥ 8 2N \ge8 2 N ≥ 8 M ≥ 1 M \ge1 M ≥ 1 N − M N-M N − M N + M N+M N + M [3 ] . Geometrically, it asks for nested squares whose L-shaped difference region has area ( N − M ) ( N + M ) (N-M)(N+M) ( N − M ) ( N + M ) A N = [ 1 , N − 3 ] A_N = [1,N-3] A N = [ 1 , N − 3 ]
C N = { M ∈ A N : N − M is prime } , D N = { M ∈ A N : N + M is prime } . C_N = \{M \in A_N : N-M \text{ is prime}\}, \qquad D_N = \{M \in A_N : N+M \text{ is prime}\}. C N = { M ∈ A N : N − M is prime } , D N = { M ∈ A N : N + M is prime } . The distinct-prime problem for 2 N 2N 2 N C N ∩ D N ≠ ∅ C_N \cap D_N \ne\varnothing C N ∩ D N = ∅ 2 N 2N 2 N R ( 2 N ) = ∣ C N ∩ D N ∣ R(2N) = |C_N \cap D_N| R ( 2 N ) = ∣ C N ∩ D N ∣ Proposition 3.2 ). Our approach embeds this centered condition in the straddling condition defining the half-difference set
E N = { Q − P 2 ∈ A N ∣ P , Q prime , 2 < P < N < Q < 2 N } . E_N = \left\{\frac{Q-P}{2} \in A_N \mathrel{\Big|} P,Q\text{ prime},\ 2 < P < N < Q < 2N\right\}. E N = { 2 Q − P ∈ A N P , Q prime , 2 < P < N < Q < 2 N } . an element M M M C N ∩ D N C_N \cap D_N C N ∩ D N ( N − M , N + M ) (N-M,N+M) ( N − M , N + M ) N N N C N ∩ D N ⊂ E N C_N \cap D_N \subset E_N C N ∩ D N ⊂ E N Lemma 3.3 ), while membership in E N E_N E N N N N M M M
Coverage of A N A_N A N ∣ A N ∖ E N ∣ ≪ D N / ( log N ) D |A_N \setminus E_N| \ll_D N/(\log N)^D ∣ A N ∖ E N ∣ ≪ D N / ( log N ) D D > 0 D > 0 D > 0 Proposition 3.11 ), which we deduce from Mikawa’s theorem on the Hardy–Littlewood asymptotic for prime pairs with almost all differences [4 ] . With explicit constants, ∣ E N ∣ > N / 760 |E_N| > N/760 ∣ E N ∣ > N /760 N ≥ 3275 N \ge3275 N ≥ 3275 Theorem 3.9 ); the proof combines the Cauchy–Schwarz inequality, an exact identity for the additive energy of the primes below and above N N N Lemma 3.7 ), Siebert’s explicit sieve bound [5 ] , and the estimates of Dusart [6 ] and of Rosser and Schoenfeld [7 ] .
Goldbach partitions up to a difference of two primes. For all large N N N ∣ C N ∩ E N ∣ = π ( N − 1 ) − 1 − O D ( N / ( log N ) D ) |C_N \cap E_N| = \pi(N-1)-1-O_D(N/(\log N)^D) ∣ C N ∩ E N ∣ = π ( N − 1 ) − 1 − O D ( N / ( log N ) D ) O D ( N / ( log N ) D ) O_D(N/(\log N)^D) O D ( N / ( log N ) D ) 3 ≤ P < N 3 \le P < N 3 ≤ P < N 2 < P ′ < N < Q ′ < 2 N 2 < P' < N < Q' < 2N 2 < P ′ < N < Q ′ < 2 N Q ′ = ( 2 N − P ) + ( P ′ − P ) Q' = (2N-P)+(P'-P) Q ′ = ( 2 N − P ) + ( P ′ − P ) 2 N = P + Q ′ 2N = P+Q' 2 N = P + Q ′ P ′ = P P'=P P ′ = P C N ∩ E N ≠ ∅ C_N \cap E_N \ne\varnothing C N ∩ E N = ∅ N ≥ 356 N \ge356 N ≥ 356 H ( N ) > 0 H(N)>0 H ( N ) > 0 H ( N ) = log 2 ( 2 N ) − ∣ A N ∖ E N ∣ H(N)=\log^2(2N)-|A_N \setminus E_N| H ( N ) = log 2 ( 2 N ) − ∣ A N ∖ E N ∣ Proposition 5.3 ).
The centering step. In contrast with E N E_N E N ∣ C N ∩ D N ∣ < ( N − 3 ) − log 2 ( 2 N ) |C_N \cap D_N| < (N-3)-\log^2(2N) ∣ C N ∩ D N ∣ < ( N − 3 ) − log 2 ( 2 N ) N ≥ 34 N \ge34 N ≥ 34 Proposition 5.6 ). The prime gap around N N N 2 N 2N 2 N E N E_N E N Lemmas 3.4 and 3.5 ); consequently H ( N ) > 0 H(N)>0 H ( N ) > 0 N N N p ′ − p < 2 log 2 ( 2 p + 2 ) + 2 p'-p < 2\log^2(2p+2)+2 p ′ − p < 2 log 2 ( 2 p + 2 ) + 2 p < p ′ p<p' p < p ′ Theorem 5.4 ), and the theorem of Ford, Green, Konyagin, Maynard and Tao on long prime gaps [8 ] shows that ∣ A N ∖ E N ∣ |A_N \setminus E_N| ∣ A N ∖ E N ∣
An explicit positive proportion of Goldbach numbers. The method of (1) shows that 2 N 2N 2 N X / 9600 X/9600 X /9600 N ≤ X N \le X N ≤ X X ≥ 10 6 X \ge10^6 X ≥ 1 0 6 Theorem 3.13 ).
Computation. For every N ≤ 10 6 N \le10^6 N ≤ 1 0 6 E N E_N E N R ( 2 N ) R(2N) R ( 2 N ) H ( N ) > 0 H(N)>0 H ( N ) > 0 ∣ E N ∣ ≥ 0.9906 N |E_N| \ge0.9906N ∣ E N ∣ ≥ 0.9906 N N ≥ 3275 N \ge3275 N ≥ 3275 R ( 2 N ) R(2N) R ( 2 N ) E N E_N E N 0.2698 0.2698 0.2698 ( 2 19 , 2 20 ] (2^{19},2^{20}] ( 2 19 , 2 20 ] § 4 ).
The symmetric reformulation [3 ] , its geometric form (§ 3 ), and the verified range (Proposition 5.7 , which combines the verification of Oliveira e Silva, Herzog and Pardi [2 ] with our computation) are the setting of these results. By Proposition 5.7 , the distinct-prime Goldbach conjecture is equivalent to C N ∩ D N ≠ ∅ C_N \cap D_N \ne\varnothing C N ∩ D N = ∅ N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18 § 2 collects the analytic number theory prerequisites used in the quantitative discussion. § 3 presents the geometric setting and the set E N E_N E N Theorems 3.9 and 3.13 . § 4 presents computational evidence. § 5 records the rigorous consequences, the open centered problem, and the small cases. § 6 summarises the results and lists open questions.
Preliminaries: Prime Distribution Results
In this section we collect the results from analytic number theory that are used in the quantitative discussion. We use explicit estimates of Dusart [6 ] and Rosser–Schoenfeld [7 ] .
Primes in Short Intervals
The following result guarantees the existence of at least one prime in every sufficiently short interval. It is the principal tool for controlling the density of primes in ( N , 2 N ) (N,2N) ( N , 2 N )
Proposition 2.1 ([6 Théorème 1.9, p. 35] ). For every real number x ≥ 3275 x \ge3275 x ≥ 3275 p p p
x < p ≤ x ( 1 + 1 2 ln 2 x ) . x < p \le x\left(1+\frac{1}{2\ln^2 x}\right). x < p ≤ x ( 1 + 2 ln 2 x 1 ) . Proposition 2.1 is derived from the following bound on consecutive primes.
Proposition 2.2 ([6 Proposition 1.10, p. 34] ). For k ≥ 463 k \ge463 k ≥ 463 p k ≥ 3299 p_k \ge3299 p k ≥ 3299 p k p_k p k p k + 1 p_{k+1} p k + 1
p k + 1 ≤ p k ( 1 + 1 2 ln 2 p k ) . p_{k+1} \le p_k\left(1+\frac{1}{2\ln^2 p_k}\right). p k + 1 ≤ p k ( 1 + 2 ln 2 p k 1 ) . The proof of Proposition 2.1 in [6 ] proceeds by using Proposition 2.2 for all x ≥ p 463 = 3299 x \ge p_{463}=3299 x ≥ p 463 = 3299 3275 ≤ x < 3299 3275 \le x < 3299 3275 ≤ x < 3299
Bounds on the Prime-Counting Function
We use the following bounds of Dusart [6 ] and of Rosser and Schoenfeld [7 ] for π ( x ) \pi(x) π ( x ) x x x
Proposition 2.3 ([6 Théorème 1.10, p. 36] ). The following inequalities hold:
x ln x ( 1 + 1 ln x ) ≤ π ( x ) for all x ≥ 599. (1) \frac{x}{\ln x}\left(1+\frac{1}{\ln x}\right)\leq\pi(x)\qquad\text{for all }x\geq599. \tag*{(1)} ln x x ( 1 + ln x 1 ) ≤ π ( x ) for all x ≥ 599. (1) π ( x ) ≤ x ln x ( 1 + 1.2762 ln x ) for all x > 1. (2) \pi(x)\leq\frac{x}{\ln x}\left(1+\frac{1.2762}{\ln x}\right)\qquad\text{for all }x>1. \tag*{(2)} π ( x ) ≤ ln x x ( 1 + ln x 1.2762 ) for all x > 1. (2) Proposition 2.4 ([7 Corollary 1, (3.5)] ). For every real number x ≥ 17 x\geq17 x ≥ 17
π ( x ) > x ln x \pi(x)>\frac{x}{\ln x} π ( x ) > ln x x Throughout this paper, log denotes the natural logarithm (i.e., log ≡ ln \log\equiv\ln log ≡ ln
Primes in Dyadic Intervals
We shall also use the following classical explicit form of Bertrand’s postulate due to Rosser and Schoenfeld [7 ] .
Proposition 2.5 ([7 Corollaries 3.8–3.9] ). For every real number x ≥ 41 / 2 x\geq41/2 x ≥ 41/2
3 x 5 log x < π ( 2 x ) − π ( x ) < 7 x 5 log x . \frac{3x}{5\log x}<\pi(2x)-\pi(x)<\frac{7x}{5\log x}. 5 log x 3 x < π ( 2 x ) − π ( x ) < 5 log x 7 x . An Elementary Inequality
Lemma 2.6. For every real number x ≥ 356 x\geq356 x ≥ 356
x log x + 2 − 2 > log 2 ( 2 x ) . \frac{x}{\log x+2}-2>\log^{2}(2x). log x + 2 x − 2 > log 2 ( 2 x ) . Proof. Let f ( x ) = x log x + 2 − 2 − log 2 ( 2 x ) f(x)=\frac{x}{\log x+2}-2-\log^{2}(2x) f ( x ) = l o g x + 2 x − 2 − log 2 ( 2 x ) x > 1 x>1 x > 1 L = log x L=\log x L = log x
f ′ ( x ) = L + 1 ( L + 2 ) 2 − 2 ( L + log 2 ) x , f'(x)=\frac{L+1}{(L+2)^{2}}-\frac{2(L+\log2)}{x}, f ′ ( x ) = ( L + 2 ) 2 L + 1 − x 2 ( L + log 2 ) , so f ′ ( x ) > 0 f'(x)>0 f ′ ( x ) > 0 x ( L + 1 ) > 2 ( L + log 2 ) ( L + 2 ) 2 x(L+1)>2(L+\log2)(L+2)^{2} x ( L + 1 ) > 2 ( L + log 2 ) ( L + 2 ) 2 log 2 < 1 \log2<1 log 2 < 1 2 ( L + log 2 ) ( L + 2 ) 2 < 2 ( L + 1 ) ( L + 2 ) 2 2(L+\log2)(L+2)^{2}<2(L+1)(L+2)^{2} 2 ( L + log 2 ) ( L + 2 ) 2 < 2 ( L + 1 ) ( L + 2 ) 2 f ′ ( x ) > 0 f'(x)>0 f ′ ( x ) > 0 x > 2 ( L + 2 ) 2 x>2(L+2)^{2} x > 2 ( L + 2 ) 2
h ( L ) : = L − log 2 − 2 log ( L + 2 ) > 0. h(L):=L-\log2-2\log(L+2)>0. h ( L ) := L − log 2 − 2 log ( L + 2 ) > 0. For L > 0 L>0 L > 0 h ′ ( L ) = 1 − 2 L + 2 > 0 h'(L)=1-\frac{2}{L+2}>0 h ′ ( L ) = 1 − L + 2 2 > 0 h h h ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) L = log 356 L=\log356 L = log 356 2 ( L + 2 ) 2 < 124.1 < 356 = e L 2(L+2)^{2}<124.1<356=e^{L} 2 ( L + 2 ) 2 < 124.1 < 356 = e L h ( log 356 ) > 0 h(\log356)>0 h ( log 356 ) > 0 h ( L ) > 0 h(L)>0 h ( L ) > 0 L ≥ log 356 L\geq\log356 L ≥ log 356 f f f [ 356 , ∞ ) [356,\infty) [ 356 , ∞ )
356 log 356 + 2 − 2 > 43.206 and log 2 ( 712 ) < 43.140 , \frac{356}{\log356+2}-2>43.206\qquad\text{and}\qquad\log^{2}(712)<43.140, log 356 + 2 356 − 2 > 43.206 and log 2 ( 712 ) < 43.140 , so f ( 356 ) > 0 f(356)>0 f ( 356 ) > 0 f ( x ) > 0 f(x)>0 f ( x ) > 0 x ≥ 356 x\geq356 x ≥ 356
Geometric Construction and Reformulation
We now develop the geometric framework used in the reformulation.
Nested Squares and Semiprime Areas
Consider a square S N S_N S N N ≥ 4 N \ge4 N ≥ 4 N 2 N^2 N 2 S N S_N S N S M S_M S M M M M 1 ≤ M ≤ N − 3 1 \le M \le N - 3 1 ≤ M ≤ N − 3 S N S_N S N S N S_N S N S M S_M S M
N 2 − M 2 = ( N − M ) ( N + M ) . N^2 - M^2 = (N - M)(N + M). N 2 − M 2 = ( N − M ) ( N + M ) . Define P = N − M P = N - M P = N − M Q = N + M Q = N + M Q = N + M M M M P P P Q Q Q M ≥ 1 M \ge1 M ≥ 1 P ≤ N − 1 P \le N - 1 P ≤ N − 1 Q ≥ N + 1 Q \ge N + 1 Q ≥ N + 1 M ≤ N − 3 M \le N - 3 M ≤ N − 3 P ≥ 3 P \ge3 P ≥ 3 3 ≤ P ≤ N − 1 3 \le P \le N - 1 3 ≤ P ≤ N − 1 Q ≥ N + 1 Q \ge N + 1 Q ≥ N + 1 P < Q P < Q P < Q M ≥ 1 M \ge1 M ≥ 1
Connection to Goldbach Partitions
The sum and difference of P P P Q Q Q
P + Q = ( N − M ) + ( N + M ) = 2 N ≥ 8 , Q − P = ( N + M ) − ( N − M ) = 2 M . \begin{aligned}
P + Q &= (N - M) + (N + M) = 2N \ge8,\\
Q - P &= (N + M) - (N - M) = 2M.
\end{aligned} P + Q Q − P = ( N − M ) + ( N + M ) = 2 N ≥ 8 , = ( N + M ) − ( N − M ) = 2 M . Since both the sum and difference are even, P P P Q Q Q P ≥ 3 P \ge3 P ≥ 3 N 2 − M 2 = P ⋅ Q N^2 - M^2 = P \cdot Q N 2 − M 2 = P ⋅ Q P P P Q Q Q
The Geometric Equivalence
Theorem 3.1 (Geometric Goldbach Variant). The following are equivalent for all N ≥ 4 N \ge4 N ≥ 4
(i) The even integer 2 N 2N 2 N
(ii) There exists M ∈ [ 1 , N − 3 ] M \in[1, N - 3] M ∈ [ 1 , N − 3 ] P = N − M P = N - M P = N − M Q = N + M Q = N + M Q = N + M
(iii) The L-shaped region between squares S N S_N S N S M S_M S M ( N − M ) ( N + M ) (N - M)(N + M) ( N − M ) ( N + M ) M ∈ [ 1 , N − 3 ] M \in[1, N - 3] M ∈ [ 1 , N − 3 ]
Proof. (i) ⇒ \Rightarrow ⇒ If 2 N = p + q 2N = p + q 2 N = p + q p < q p < q p < q M = ( q − p ) / 2 M = (q - p)/2 M = ( q − p ) /2 p p p q q q 2 N ≥ 8 2N \ge8 2 N ≥ 8 ( q + p ) / 2 = N (q + p)/2 = N ( q + p ) /2 = N ( q − p ) / 2 = M (q - p)/2 = M ( q − p ) /2 = M P = N − M = p P = N - M = p P = N − M = p Q = N + M = q Q = N + M = q Q = N + M = q M ∈ [ 1 , N − 3 ] M \in[1, N - 3] M ∈ [ 1 , N − 3 ] q − p ≥ 2 q - p \ge2 q − p ≥ 2 M ≥ 1 M \ge1 M ≥ 1 M ≤ N − 3 M \le N - 3 M ≤ N − 3 p ≥ 3 p \ge3 p ≥ 3 p p p (ii) ⇒ \Rightarrow ⇒ Immediate, as N 2 − M 2 = ( N − M ) ( N + M ) N^2 - M^2 = (N - M)(N + M) N 2 − M 2 = ( N − M ) ( N + M ) (iii) ⇒ \Rightarrow ⇒ If N 2 − M 2 = ( N − M ) ( N + M ) N^2 - M^2 = (N - M)(N + M) N 2 − M 2 = ( N − M ) ( N + M ) M ∈ [ 1 , N − 3 ] M \in[1, N - 3] M ∈ [ 1 , N − 3 ] N − M N - M N − M N + M N + M N + M 2 N 2N 2 N M ≥ 1 M \ge1 M ≥ 1
Figure 1. The geometric construction for N = 5 N = 5 N = 5 M = 2 M = 2 M = 2 N 2 − M 2 = 25 − 4 = 21 = 3 × 7 N^2 - M^2 = 25 - 4 = 21 = 3 \times7 N 2 − M 2 = 25 − 4 = 21 = 3 × 7 P = 3 P = 3 P = 3 Q = 7 Q = 7 Q = 7 2 N = 10 2N = 10 2 N = 10 10 = 3 + 7 10 = 3 + 7 10 = 3 + 7
Reformulation as a Centered Set Intersection Problem
Let
A N = { 1 , 2 , … , N − 3 } A_N = \{1, 2, \ldots, N - 3\} A N = { 1 , 2 , … , N − 3 } be the admissible range for M M M N ≥ 4 N \ge4 N ≥ 4 A N A_N A N
Lower-prime candidate set: C N = { M ∈ A N ∣ N − M is prime } C_N = \{M \in A_N \mid N - M \text{ is prime}\} C N = { M ∈ A N ∣ N − M is prime }
Upper-prime candidate set: D N = { M ∈ A N ∣ N + M is prime } D_N = \{M \in A_N \mid N + M \text{ is prime}\} D N = { M ∈ A N ∣ N + M is prime }
Both centered sets lie in a single parity class. Indeed, if N − M N - M N − M N + M N + M N + M M M M N N N
C N , D N ⊆ B N : = { M ∈ A N ∣ M ≢ N ( m o d 2 ) } . C_N, D_N \subseteq B_N := \{M \in A_N \mid M \not\equiv N \pmod{2}\}. C N , D N ⊆ B N := { M ∈ A N ∣ M ≡ N ( mod 2 )} . Proposition 3.2 (Centered intersection criterion). The distinct-prime Goldbach conjecture holds for the even integer 2 N 2N 2 N N ≥ 4 N \ge4 N ≥ 4 C N ∩ D N ≠ ∅ C_N \cap D_N \ne\varnothing C N ∩ D N = ∅
Proof. (⇒ \Rightarrow ⇒ 2 N = p + q 2N = p + q 2 N = p + q p < q p < q p < q M = ( q − p ) / 2 M = (q - p)/2 M = ( q − p ) /2 Theorem 3.1 , M ∈ A N M \in A_N M ∈ A N p = N − M p = N - M p = N − M q = N + M q = N + M q = N + M M ∈ C N ∩ D N M \in C_N \cap D_N M ∈ C N ∩ D N
(⇐ \Leftarrow ⇐ M ∈ C N ∩ D N M \in C_N \cap D_N M ∈ C N ∩ D N N − M N - M N − M C N C_N C N N + M N + M N + M D N D_N D N 2 N 2N 2 N M ≥ 1 M \ge1 M ≥ 1
The Half-Difference Diagnostic Set
It is useful computationally to compare the centered sets with a set of half-differences arising from straddling prime pairs. Define
E N = { Q − P 2 ∈ A N ∣ 2 < P < N < Q < 2 N , P , Q prime } . E_N = \left\{\frac{Q-P}{2} \in A_N \mid2 < P < N < Q < 2N,\ P,Q\text{ prime}\right\}. E N = { 2 Q − P ∈ A N ∣ 2 < P < N < Q < 2 N , P , Q prime } . The restriction Q − P 2 ∈ A N \frac{Q-P}{2} \in A_N 2 Q − P ∈ A N E N E_N E N { 1 , … , N − 3 } \{1,\ldots,N-3\} { 1 , … , N − 3 } E N E_N E N N N N D N D_N D N N + M N+M N + M C N ∩ D N C_N \cap D_N C N ∩ D N
Lemma 3.3. For every N ≥ 4 N \ge4 N ≥ 4
C N ∩ D N ⊆ E N . C_N \cap D_N \subseteq E_N. C N ∩ D N ⊆ E N . Consequently, C N ∩ E N ∩ D N = C N ∩ D N C_N \cap E_N \cap D_N = C_N \cap D_N C N ∩ E N ∩ D N = C N ∩ D N
Proof. Let M ∈ C N ∩ D N M \in C_N \cap D_N M ∈ C N ∩ D N P = N − M P=N-M P = N − M Q = N + M Q=N+M Q = N + M M ∈ A N M \in A_N M ∈ A N P ≥ 3 > 2 P \ge3 > 2 P ≥ 3 > 2 P < N < Q P<N<Q P < N < Q Q ≤ 2 N − 3 < 2 N Q \le2N-3<2N Q ≤ 2 N − 3 < 2 N M = Q − P 2 ∈ A N M=\frac{Q-P}{2}\in A_N M = 2 Q − P ∈ A N M ∈ E N M\in E_N M ∈ E N C N C_N C N
Lemma 3.4 (Gap obstruction). Let N ≥ 4 N \ge4 N ≥ 4 p − p^- p − N N N p + p^+ p + N N N M ∈ E N M \in E_N M ∈ E N 2 M ≥ p + − p − 2M \ge p^+ - p^- 2 M ≥ p + − p − M ∈ A N M \in A_N M ∈ A N M < ( p + − p − ) / 2 M < (p^+ - p^-)/2 M < ( p + − p − ) /2 A N ∖ E N A_N \setminus E_N A N ∖ E N
Proof. If M = ( Q − P ) / 2 M=(Q-P)/2 M = ( Q − P ) /2 P < N < Q P<N<Q P < N < Q P ≤ p − P\le p^- P ≤ p − Q ≥ p + Q\ge p^+ Q ≥ p + 2 M = Q − P ≥ p + − p − 2M=Q-P\ge p^+-p^- 2 M = Q − P ≥ p + − p −
Lemma 3.5 (Upper gap obstruction). Let N ≥ 4 N \ge4 N ≥ 4 q q q 2 N 2N 2 N M ∈ E N M \in E_N M ∈ E N 2 M ≤ q − 3 2M \le q-3 2 M ≤ q − 3 M ∈ A N M \in A_N M ∈ A N M > ( q − 3 ) / 2 M>(q-3)/2 M > ( q − 3 ) /2 A N ∖ E N A_N\setminus E_N A N ∖ E N q ≤ 2 N − 3 q\le2N-3 q ≤ 2 N − 3 ( 2 N − 3 − q ) / 2 (2N-3-q)/2 ( 2 N − 3 − q ) /2
Proof. If M = ( Q − P ) / 2 M=(Q-P)/2 M = ( Q − P ) /2 2 < P 2<P 2 < P Q < 2 N Q<2N Q < 2 N P ≥ 3 P\ge3 P ≥ 3 Q ≤ q Q\le q Q ≤ q 2 M ≤ q − 3 2M\le q-3 2 M ≤ q − 3 M ∈ A N M\in A_N M ∈ A N M > ( q − 3 ) / 2 M>(q-3)/2 M > ( q − 3 ) /2 M = ( q − 3 ) / 2 + 1 , … , N − 3 M=(q-3)/2+1,\ldots,N-3 M = ( q − 3 ) /2 + 1 , … , N − 3 N − 3 − ( q − 3 ) / 2 = ( 2 N − 3 − q ) / 2 N-3-(q-3)/2=(2N-3-q)/2 N − 3 − ( q − 3 ) /2 = ( 2 N − 3 − q ) /2
Nearby Even Numbers Represented by E N E_N E N
Each element of E N E_N E N M ∈ E N M\in E_N M ∈ E N P , Q P,Q P , Q
2 < P < N < Q < 2 N , M = Q − P 2 , 2<P<N<Q<2N,\qquad M=\frac{Q-P}{2}, 2 < P < N < Q < 2 N , M = 2 Q − P , and the even number
has a Goldbach partition with half-difference M M M S = P + Q S=P+Q S = P + Q Q − P 2 ∈ A N \frac{Q-P}{2}\in A_N 2 Q − P ∈ A N E N E_N E N S S S
N + 4 ≤ S ≤ 3 N − 2 , N+4\le S\le3N-2, N + 4 ≤ S ≤ 3 N − 2 , with endpoint adjustments forced by parity and by the primality of P P P Q Q Q Q = 2 N − 1 Q=2N-1 Q = 2 N − 1 3 N − 4 3N-4 3 N − 4 N N N 2 N 2N 2 N S = 2 N S=2N S = 2 N
2 N = ( N − M ) + ( N + M ) . 2N=(N-M)+(N+M). 2 N = ( N − M ) + ( N + M ) . Lower and Upper Primes
Let
L N = { P ∣ 3 ≤ P < N , P prime } , a N = ∣ L N ∣ = π ( N − 1 ) − 1 , L_N=\{P\mid3\le P<N,\ P\text{ prime}\},\qquad a_N=|L_N|=\pi(N-1)-1, L N = { P ∣ 3 ≤ P < N , P prime } , a N = ∣ L N ∣ = π ( N − 1 ) − 1 , and let
U N = { Q ∣ N < Q ≤ 2 N − 3 , Q prime } = { Q 1 > Q 2 > ⋯ > Q r } . U_N=\{Q\mid N<Q\le2N-3,\ Q\text{ prime}\}=\{Q_1>Q_2>\cdots>Q_r\}. U N = { Q ∣ N < Q ≤ 2 N − 3 , Q prime } = { Q 1 > Q 2 > ⋯ > Q r } . If 2 N − 3 2N-3 2 N − 3 2 N = 3 + ( 2 N − 3 ) 2N=3+(2N-3) 2 N = 3 + ( 2 N − 3 ) M = N − 3 M=N-3 M = N − 3 r r r π ( 2 N ) − π ( N ) \pi(2N)-\pi(N) π ( 2 N ) − π ( N ) π ( 2 N − 3 ) − π ( N ) \pi(2N-3)-\pi(N) π ( 2 N − 3 ) − π ( N ) 2 N − 1 2N-1 2 N − 1
π ( 2 N ) − π ( N ) − 1 ≤ r ≤ π ( 2 N ) − π ( N ) . \pi(2N)-\pi(N)-1\le r\le\pi(2N)-\pi(N). π ( 2 N ) − π ( N ) − 1 ≤ r ≤ π ( 2 N ) − π ( N ) . Proposition 2.5 therefore gives, for N ≥ 41 / 2 N \ge41/2 N ≥ 41/2
3 N 5 log N − 1 < r < 7 N 5 log N . \frac{3N}{5\log N}-1<r<\frac{7N}{5\log N}. 5 log N 3 N − 1 < r < 5 log N 7 N . For P ∈ L N P\in L_N P ∈ L N Q ∈ U N Q\in U_N Q ∈ U N 3 ≤ P < N < Q ≤ 2 N − 3 3\le P<N<Q\le2N-3 3 ≤ P < N < Q ≤ 2 N − 3 1 ≤ ( Q − P ) / 2 ≤ N − 3 1\le(Q-P)/2\le N-3 1 ≤ ( Q − P ) /2 ≤ N − 3 ( Q − P ) / 2 ∈ E N (Q-P)/2\in E_N ( Q − P ) /2 ∈ E N ( P , Q ) ∈ L N × U N (P,Q)\in L_N\times U_N ( P , Q ) ∈ L N × U N d > 0 d>0 d > 0
ρ L ( d ) = # { ( P , P ′ ) ∈ L N 2 ∣ P ′ − P = d } , ρ U ( d ) = # { ( i , j ) ∣ i < j , Q i − Q j = d } , \rho_L(d)=\#\{(P,P')\in L_N^2\mid P'-P=d\},\qquad\rho_U(d)=\#\{(i,j)\mid i<j,\ Q_i-Q_j=d\}, ρ L ( d ) = # {( P , P ′ ) ∈ L N 2 ∣ P ′ − P = d } , ρ U ( d ) = # {( i , j ) ∣ i < j , Q i − Q j = d } , and the additive energy
C N = ∑ d > 0 ρ L ( d ) ρ U ( d ) , C_N=\sum_{d>0}\rho_L(d)\rho_U(d), C N = d > 0 ∑ ρ L ( d ) ρ U ( d ) , which counts the quadruples ( P , P ′ , Q i , Q j ) (P,P',Q_i,Q_j) ( P , P ′ , Q i , Q j ) P , P ′ ∈ L N P,P'\in L_N P , P ′ ∈ L N i < j i<j i < j P ′ − P = Q i − Q j P'-P=Q_i-Q_j P ′ − P = Q i − Q j
A Linear Lower Bound for the Half-Difference Set
The additive energy C N C_N C N § 3.7 controls ∣ E N ∣ |E_N| ∣ E N ∣ [5 ] for primes p p p a p + b ap+b a p + b 2 ∣ a b 2\mid ab 2 ∣ ab a = 1 a=1 a = 1 b = h b=h b = h
Proposition 3.6 ([5 ] ). Let
ω = ∏ p > 2 ( 1 − 1 ( p − 1 ) 2 ) , κ ( h ) = ∏ p ∣ h , p > 2 p − 1 p − 2 . \omega=\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right),\qquad\kappa(h)=\prod_{p\mid h,\ p>2}\frac{p-1}{p-2}. ω = p > 2 ∏ ( 1 − ( p − 1 ) 2 1 ) , κ ( h ) = p ∣ h , p > 2 ∏ p − 2 p − 1 . For every real number x > 1 x>1 x > 1 h ≠ 0 h\ne0 h = 0
# { p ≤ x ∣ p and p + h are prime } ≤ 16 ω κ ( h ) x log 2 x . \#\{p\le x\mid p\text{ and }p+h\text{ are prime}\}\le16\omega\kappa(h)\frac{x}{\log^2x}. # { p ≤ x ∣ p and p + h are prime } ≤ 16 ω κ ( h ) log 2 x x . For M ∈ A N M\in A_N M ∈ A N
ν N ( M ) = # { ( P , Q ) ∈ L N × U N ∣ Q − P = 2 M } . \nu_N(M)=\#\{(P,Q)\in L_N\times U_N\mid Q-P=2M\}. ν N ( M ) = # {( P , Q ) ∈ L N × U N ∣ Q − P = 2 M } . Every pair ( P , Q ) ∈ L N × U N (P,Q)\in L_N\times U_N ( P , Q ) ∈ L N × U N ( Q − P ) / 2 ∈ E N (Q-P)/2\in E_N ( Q − P ) /2 ∈ E N § 3.7 . Hence ν N \nu_N ν N E N E_N E N
∑ M ∈ E N ν N ( M ) = a N r . \sum_{M\in E_N}\nu_N(M)=a_Nr. M ∈ E N ∑ ν N ( M ) = a N r . Lemma 3.7 (Energy identity). For every N ≥ 4 N\ge4 N ≥ 4
∑ M ∈ A N ν N ( M ) 2 = a N r + 2 C N . \sum_{M\in A_N}\nu_N(M)^2=a_Nr+2C_N. M ∈ A N ∑ ν N ( M ) 2 = a N r + 2 C N . Proof. The left-hand side counts the quadruples ( P , Q , P ′ , Q ′ ) ∈ L N × U N × L N × U N (P,Q,P',Q')\in L_N\times U_N\times L_N\times U_N ( P , Q , P ′ , Q ′ ) ∈ L N × U N × L N × U N Q − P = Q ′ − P ′ Q-P=Q'-P' Q − P = Q ′ − P ′ P ′ − P = Q ′ − Q = : h P'-P=Q'-Q=:h P ′ − P = Q ′ − Q =: h h = 0 h=0 h = 0 P = P ′ P=P' P = P ′ Q = Q ′ Q=Q' Q = Q ′ a N r a_Nr a N r h > 0 h>0 h > 0 ρ L ( h ) ρ U ( h ) \rho_L(h)\rho_U(h) ρ L ( h ) ρ U ( h ) ( P , Q ) (P,Q) ( P , Q ) ( P ′ , Q ′ ) (P',Q') ( P ′ , Q ′ ) h h h − h -h − h
Lemma 3.8 (Energy bound). Let
G = ∏ p > 2 ( 1 + 2 p − 3 p ( p − 2 ) 2 ) . G=\prod_{p>2}\left(1+\frac{2p-3}{p(p-2)^2}\right). G = p > 2 ∏ ( 1 + p ( p − 2 ) 2 2 p − 3 ) . For every N ≥ 4 N\ge4 N ≥ 4
C N ≤ 256 ω 2 G N 3 log 4 N . C_N\le256\omega^2G\frac{N^3}{\log^4N}. C N ≤ 256 ω 2 G log 4 N N 3 . Moreover, ω < 0.6614 \omega<0.6614 ω < 0.6614 G < 2.658 G<2.658 G < 2.658
Proof. All primes in L N ∪ U N L_N \cup U_N L N ∪ U N ρ L ( h ) = 0 \rho_L(h)=0 ρ L ( h ) = 0 h h h 2 ≤ h ≤ N − 4 2 \le h \le N-4 2 ≤ h ≤ N − 4 h h h Proposition 3.6 with x = N x=N x = N x = 2 N x=2N x = 2 N
ρ L ( h ) ≤ 16 ω κ ( h ) N log 2 N , ρ U ( h ) ≤ 16 ω κ ( h ) 2 N log 2 ( 2 N ) ≤ 32 ω κ ( h ) N log 2 N , \rho_L(h) \le16\omega\kappa(h)\frac{N}{\log^2 N}, \qquad\rho_U(h) \le16\omega\kappa(h)\frac{2N}{\log^2(2N)} \le32\omega\kappa(h)\frac{N}{\log^2 N}, ρ L ( h ) ≤ 16 ω κ ( h ) log 2 N N , ρ U ( h ) ≤ 16 ω κ ( h ) log 2 ( 2 N ) 2 N ≤ 32 ω κ ( h ) log 2 N N , since every pair counted by ρ U ( h ) \rho_U(h) ρ U ( h ) Q < Q + h ≤ 2 N Q<Q+h\le2N Q < Q + h ≤ 2 N
C N ≤ 512 ω 2 N 2 log 4 N ∑ 1 ≤ k ≤ N / 2 κ ( 2 k ) 2 . C_N \le512\omega^2\frac{N^2}{\log^4 N}\sum_{1\le k\le N/2}\kappa(2k)^2. C N ≤ 512 ω 2 log 4 N N 2 1 ≤ k ≤ N /2 ∑ κ ( 2 k ) 2 . Since κ \kappa κ κ ( 2 k ) = κ ( k ) \kappa(2k)=\kappa(k) κ ( 2 k ) = κ ( k ) κ 2 \kappa^2 κ 2 κ ( k ) 2 = ∑ d ∣ k g ( d ) \kappa(k)^2=\sum_{d\mid k}g(d) κ ( k ) 2 = ∑ d ∣ k g ( d ) g g g
g ( p ) = ( p − 1 p − 2 ) 2 − 1 = 2 p − 3 ( p − 2 ) 2 ( p > 2 ) . g(p)=\left(\frac{p-1}{p-2}\right)^2-1=\frac{2p-3}{(p-2)^2}\qquad(p>2). g ( p ) = ( p − 2 p − 1 ) 2 − 1 = ( p − 2 ) 2 2 p − 3 ( p > 2 ) . Since g ≥ 0 g\ge0 g ≥ 0 K ≥ 1 K\ge1 K ≥ 1
∑ k ≤ K κ ( k ) 2 = ∑ d ≤ K g ( d ) ⌊ K d ⌋ ≤ K ∑ d ≥ 1 g ( d ) d = K G . \sum_{k\le K}\kappa(k)^2=\sum_{d\le K}g(d)\left\lfloor\frac{K}{d}\right\rfloor\le K\sum_{d\ge1}\frac{g(d)}{d}=KG. k ≤ K ∑ κ ( k ) 2 = d ≤ K ∑ g ( d ) ⌊ d K ⌋ ≤ K d ≥ 1 ∑ d g ( d ) = K G . Taking K = N / 2 K=N/2 K = N /2 C N C_N C N
Every factor of ω \omega ω ( 0 , 1 ) (0,1) ( 0 , 1 ) ω \omega ω 3 ≤ p ≤ 97 3\le p\le97 3 ≤ p ≤ 97 0.66138 0.66138 0.66138 G G G 3 ≤ p ≤ 97 3\le p\le97 3 ≤ p ≤ 97 2.63013 2.63013 2.63013 p ≥ 101 p\ge101 p ≥ 101 2 p − 3 p ( p − 2 ) 2 < 2 ( p − 2 ) 2 \frac{2p-3}{p(p-2)^2}<\frac{2}{(p-2)^2} p ( p − 2 ) 2 2 p − 3 < ( p − 2 ) 2 2 p − 2 p-2 p − 2 log ( 1 + y ) ≤ y \log(1+y)\le y log ( 1 + y ) ≤ y
log ∏ p ≥ 101 ( 1 + 2 p − 3 p ( p − 2 ) 2 ) < 2 ∑ m ≥ 99 m o d d 1 m 2 ≤ 2 ( 1 99 2 + 1 2 ∫ 99 ∞ d t t 2 ) = 2 ( 1 9801 + 1 198 ) < 0.01031. \begin{aligned}
\log\prod_{p\ge101}\left(1+\frac{2p-3}{p(p-2)^2}\right)<2\sum_{\substack{m\ge99\\m\ \mathrm{odd}}}\frac{1}{m^2}\le2\left(\frac{1}{99^2}+\frac{1}{2}\int_{99}^{\infty}\frac{dt}{t^2}\right) \\
&=2\left(\frac{1}{9801}+\frac{1}{198}\right)<0.01031.
\end{aligned} log p ≥ 101 ∏ ( 1 + p ( p − 2 ) 2 2 p − 3 ) < 2 m ≥ 99 m odd ∑ m 2 1 ≤ 2 ( 9 9 2 1 + 2 1 ∫ 99 ∞ t 2 d t ) = 2 ( 9801 1 + 198 1 ) < 0.01031. Hence G < 2.63013 e 0.01031 < 2.658 G<2.63013e^{0.01031}<2.658 G < 2.63013 e 0.01031 < 2.658
Theorem 3.9 (Linear lower bound). For every integer N ≥ 3275 N\ge3275 N ≥ 3275
∣ E N ∣ > N 760 . |E_N|>\frac{N}{760}. ∣ E N ∣ > 760 N . Proof. Write L = log N L=\log N L = log N λ = log 2 \lambda=\log2 λ = log 2 Lemma 3.7 ,
( a N r ) 2 = ( ∑ M ∈ E N ν N ( M ) ) 2 ≤ ∣ E N ∣ ∑ M ∈ A N ν N ( M ) 2 = ∣ E N ∣ ( a N r + 2 C N ) . (a_N r)^2=\left(\sum_{M\in E_N}\nu_N(M)\right)^2\le|E_N|\sum_{M\in A_N}\nu_N(M)^2=|E_N|(a_Nr+2C_N). ( a N r ) 2 = ( M ∈ E N ∑ ν N ( M ) ) 2 ≤ ∣ E N ∣ M ∈ A N ∑ ν N ( M ) 2 = ∣ E N ∣ ( a N r + 2 C N ) . Lower bound for a N r a_Nr a N r By Proposition 2.4 with x = N − 1 ≥ 17 x=N-1\ge17 x = N − 1 ≥ 17 a N = π ( N − 1 ) − 1 > N − 1 L − 1 a_N=\pi(N-1)-1>\frac{N-1}{L}-1 a N = π ( N − 1 ) − 1 > L N − 1 − 1
a N > N L A ( N ) , A ( N ) = 1 − 1 + L N . a_N>\frac{N}{L}\mathcal{A}(N),\qquad\mathcal{A}(N)=1-\frac{1+L}{N}. a N > L N A ( N ) , A ( N ) = 1 − N 1 + L . As shown in § 3.7 , r ≥ π ( 2 N ) − π ( N ) − 1 r\ge\pi(2N)-\pi(N)-1 r ≥ π ( 2 N ) − π ( N ) − 1 Proposition 2.3 (1) with x = 2 N ≥ 599 x=2N\ge599 x = 2 N ≥ 599 Proposition 2.3 (2) with x = N x=N x = N
r ≥ 2 N L + λ ( 1 + 1 L + λ ) − N L ( 1 + 1.2762 L ) − 1 = N L B ( N ) , r\ge\frac{2N}{L+\lambda}\left(1+\frac{1}{L+\lambda}\right)-\frac{N}{L}\left(1+\frac{1.2762}{L}\right)-1=\frac{N}{L}\mathcal{B}(N), r ≥ L + λ 2 N ( 1 + L + λ 1 ) − L N ( 1 + L 1.2762 ) − 1 = L N B ( N ) , where, using 2 L L + λ = 2 − 2 λ L + λ \frac{2L}{L+\lambda}=2-\frac{2\lambda}{L+\lambda} L + λ 2 L = 2 − L + λ 2 λ
B ( N ) = 1 − 2 λ L + λ + 2 L ( L + λ ) 2 − 1.2762 L − L N . \mathcal{B}(N)=1-\frac{2\lambda}{L+\lambda}+\frac{2L}{(L+\lambda)^2}-\frac{1.2762}{L}-\frac{L}{N}. B ( N ) = 1 − L + λ 2 λ + ( L + λ ) 2 2 L − L 1.2762 − N L . Both A \mathcal{A} A B \mathcal{B} B N N N N ≥ 3275 N \ge3275 N ≥ 3275 A \mathcal{A} A ( 1 + log x ) / x (1+\log x)/x ( 1 + log x ) / x x > 1 x>1 x > 1 B \mathcal{B} B − L / N -L/N − L / N N > e N>e N > e L L L
2 λ ( λ + 1 ) − ( 2 − 2 λ ) L ( L + λ ) 3 + 1.2762 L 2 > − ( 2 − 2 λ ) L L 3 + 1.2762 L 2 = 1.2762 − ( 2 − 2 λ ) L 2 > 0 , \frac{2\lambda(\lambda+1)-(2-2\lambda)L}{(L+\lambda)^3}+\frac{1.2762}{L^2}>-\frac{(2-2\lambda)L}{L^3}+\frac{1.2762}{L^2}
=\frac{1.2762-(2-2\lambda)}{L^2}>0, ( L + λ ) 3 2 λ ( λ + 1 ) − ( 2 − 2 λ ) L + L 2 1.2762 > − L 3 ( 2 − 2 λ ) L + L 2 1.2762 = L 2 1.2762 − ( 2 − 2 λ ) > 0 , since 2 − 2 λ < 0.614 2-2\lambda<0.614 2 − 2 λ < 0.614 L ≥ log 3275 L\ge\log3275 L ≥ log 3275 N = 3275 N=3275 N = 3275 A > 0.99722 \mathcal{A}>0.99722 A > 0.99722 B > 0.89174 \mathcal{B}>0.89174 B > 0.89174 N ≥ 3275 N\ge3275 N ≥ 3275
a N r > α N 2 L 2 , α = 0.8892. a_N r>\alpha\frac{N^2}{L^2},\qquad\alpha=0.8892. a N r > α L 2 N 2 , α = 0.8892. Conclusion. The function t ↦ t 2 / ( t + K ) t\mapsto t^2/(t+K) t ↦ t 2 / ( t + K ) t > 0 t>0 t > 0 K > 0 K>0 K > 0 X = α N 2 / L 2 X=\alpha N^2/L^2 X = α N 2 / L 2 Lemma 3.8 , 2 C N ≤ K = β N 3 / L 4 2C_N\le K=\beta N^3/L^4 2 C N ≤ K = β N 3 / L 4 β = 512 ⋅ 0.6614 2 ⋅ 2.658 < 595.33 \beta=512\cdot0.6614^2\cdot2.658<595.33 β = 512 ⋅ 0.661 4 2 ⋅ 2.658 < 595.33
∣ E N ∣ ≥ ( a N r ) 2 a N r + 2 C N > X 2 X + K = α 2 N β ( 1 + δ N ) , δ N = X K = α L 2 β N . |E_N|\ge\frac{(a_Nr)^2}{a_Nr+2C_N}>\frac{X^2}{X+K}=\frac{\alpha^2N}{\beta(1+\delta_N)},\qquad\delta_N=\frac{X}{K}=\frac{\alpha L^2}{\beta N}. ∣ E N ∣ ≥ a N r + 2 C N ( a N r ) 2 > X + K X 2 = β ( 1 + δ N ) α 2 N , δ N = K X = β N α L 2 . The function t ↦ log 2 t / t t\mapsto\log^2t/t t ↦ log 2 t / t t > e 2 t>e^2 t > e 2 δ N ≤ δ 3275 < 3 ⋅ 10 − 5 \delta_N\le\delta_{3275}<3\cdot10^{-5} δ N ≤ δ 3275 < 3 ⋅ 1 0 − 5
∣ E N ∣ > 0.8892 2 595.33 ⋅ 1.00003 N > 0.001328 N > N 760 . |E_N|>\frac{0.8892^2}{595.33\cdot1.00003}N>0.001328N>\frac{N}{760}. ∣ E N ∣ > 595.33 ⋅ 1.00003 0.889 2 2 N > 0.001328 N > 760 N . □
By definition, M ∈ E N M\in E_N M ∈ E N p p p max { 2 , N − 2 M } < p < N \max\{2,N-2M\}<p<N max { 2 , N − 2 M } < p < N p + 2 M p+2M p + 2 M 2 N 2N 2 N 2 k 2k 2 k p ≤ y p\le y p ≤ y p + 2 k p+2k p + 2 k 2 ω κ ( 2 k ) y / log 2 y 2\omega\kappa(2k)y/\log^2 y 2 ω κ ( 2 k ) y / log 2 y [9 ] proved it for almost all k ≤ x k\le x k ≤ x y ∈ [ 2 x , x 8 / 5 − ε ] y\in[2x,x^{8/5-\varepsilon}] y ∈ [ 2 x , x 8/5 − ε ] [10 ] . Mikawa [4 ] extended the range of y y y [11 ] proved the asymptotic for all but O ( H ( log N ) − B ) O(H(\log N)^{-B}) O ( H ( log N ) − B ) 2 k 2k 2 k k k k H H H N 1 / 2 + ε ≤ H ≤ N / 4 N^{1/2+\varepsilon}\le H\le N/4 N 1/2 + ε ≤ H ≤ N /4 [12 ] proved that ∑ X < n ≤ 2 X Λ ( n ) Λ ( n + h ) \sum_{X<n\le2X}\Lambda(n)\Lambda(n+h) ∑ X < n ≤ 2 X Λ ( n ) Λ ( n + h ) O ( H ( log X ) − A ) O(H(\log X)^{-A}) O ( H ( log X ) − A ) h h h ∣ h − h 0 ∣ ≤ H |h-h_0|\le H ∣ h − h 0 ∣ ≤ H X 8 / 33 + ε ≤ H ≤ X 1 − ε X^{8/33+\varepsilon}\le H\le X^{1-\varepsilon} X 8/33 + ε ≤ H ≤ X 1 − ε 0 ≤ h 0 ≤ X 1 − ε 0\le h_0\le X^{1-\varepsilon} 0 ≤ h 0 ≤ X 1 − ε
Let
Ψ ( y , 2 k ) = ∑ 2 k < n ≤ y Λ ( n ) Λ ( n − 2 k ) , \Psi(y,2k)=\sum_{2k<n\le y}\Lambda(n)\Lambda(n-2k), Ψ ( y , 2 k ) = 2 k < n ≤ y ∑ Λ ( n ) Λ ( n − 2 k ) , where Λ \Lambda Λ S ( 2 k ) = 2 ω κ ( 2 k ) \mathfrak{S}(2k)=2\omega\kappa(2k) S ( 2 k ) = 2 ω κ ( 2 k ) S ( 2 k ) ≥ 2 ω \mathfrak{S}(2k)\ge2\omega S ( 2 k ) ≥ 2 ω
Proposition 3.10 ([4 ] ). Let ε , A , B > 0 \varepsilon,A,B>0 ε , A , B > 0 x x x y y y 2 x ≤ y ≤ x 3 − ε 2x\le y\le x^{3-\varepsilon} 2 x ≤ y ≤ x 3 − ε O ( x ( log x ) − A ) O(x(\log x)^{-A}) O ( x ( log x ) − A ) k k k 1 ≤ k ≤ x 1\le k\le x 1 ≤ k ≤ x
Ψ ( y , 2 k ) = S ( 2 k ) ( y − 2 k ) + O ( y ( log y ) B ) . \Psi(y,2k)=\mathfrak{S}(2k)(y-2k)+O\left(\frac{y}{(\log y)^B}\right). Ψ ( y , 2 k ) = S ( 2 k ) ( y − 2 k ) + O ( ( log y ) B y ) . where the implied constants depend only on ε \varepsilon ε A A A
We use Proposition 3.10 only for 2 x ≤ y ≤ 4 x 2x\le y\le4x 2 x ≤ y ≤ 4 x
Proposition 3.11 (Almost all half-differences). For every D > 0 D>0 D > 0
∣ A N ∖ E N ∣ ≪ D N ( log N ) D . |A_N\setminus E_N|\ll_D\frac{N}{(\log N)^D}. ∣ A N ∖ E N ∣ ≪ D ( log N ) D N . In particular, ∣ E N ∣ = N − o ( N ) |E_N|=N-o(N) ∣ E N ∣ = N − o ( N )
Proof. Write L = log N L=\log N L = log N a = ⌈ N / L D ⌉ a=\lceil N/L^{D}\rceil a = ⌈ N / L D ⌉ Proposition 3.10 with ε = 1 / 2 \varepsilon=1/2 ε = 1/2 A = D A=D A = D B = D + 1 B=D+1 B = D + 1 Ψ ( y , 2 k ) \Psi(y,2k) Ψ ( y , 2 k ) y ≤ 2 N y\le2N y ≤ 2 N n n n n − 2 k n-2k n − 2 k p j p^{j} p j j ≥ 2 j\ge2 j ≥ 2 O ( N L 3 ) O(\sqrt{N}L^{3}) O ( N L 3 ) O ( N L ) O(\sqrt{N}L) O ( N L ) 2 N 2N 2 N L 2 + O ( L ) L^{2}+O(L) L 2 + O ( L )
The range a < k ≤ N / 2 a<k\le N/2 a < k ≤ N /2 Take x = N / 2 x=N/2 x = N /2 y 1 = N y_{1}=N y 1 = N y 2 = N + 2 a y_{2}=N+2a y 2 = N + 2 a 2 x ≤ y 1 < y 2 ≤ 4 x 2x\le y_{1}<y_{2}\le4x 2 x ≤ y 1 < y 2 ≤ 4 x O ( N L − D ) O(NL^{-D}) O ( N L − D ) k ≤ x k\le x k ≤ x Proposition 3.10 applies to both values, and
Ψ ( y 2 , 2 k ) − Ψ ( y 1 , 2 k ) = ∑ N < n ≤ N + 2 a Λ ( n ) Λ ( n − 2 k ) = 2 a S ( 2 k ) + O ( N L D + 1 ) ≥ 4 ω a − O ( N L D + 1 ) , \begin{aligned}
\Psi(y_{2},2k)-\Psi(y_{1},2k)&=\sum_{N<n\le N+2a}\Lambda(n)\Lambda(n-2k)\\
&=2a\,\mathfrak{S}(2k)+O\left(\frac{N}{L^{D+1}}\right)\ge4\omega a-O\left(\frac{N}{L^{D+1}}\right),
\end{aligned} Ψ ( y 2 , 2 k ) − Ψ ( y 1 , 2 k ) = N < n ≤ N + 2 a ∑ Λ ( n ) Λ ( n − 2 k ) = 2 a S ( 2 k ) + O ( L D + 1 N ) ≥ 4 ω a − O ( L D + 1 N ) , which exceeds the prime-power contribution O ( N L 3 ) O(\sqrt{N}L^{3}) O ( N L 3 ) N N N n n n N < n ≤ N + 2 a N<n\le N+2a N < n ≤ N + 2 a Q = n Q=n Q = n P = n − 2 k P=n-2k P = n − 2 k N < Q < 2 N N<Q<2N N < Q < 2 N P ≤ N + 2 a − 2 k < N P\le N+2a-2k<N P ≤ N + 2 a − 2 k < N k > a k>a k > a P P P Q Q Q 2 k 2k 2 k P > 2 P>2 P > 2 k ∈ E N k\in E_N k ∈ E N
The range N / 2 < k ≤ N − 3 − a N/2<k\le N-3-a N /2 < k ≤ N − 3 − a Take x = N − 1 x=N-1 x = N − 1 y = 2 N − 1 y=2N-1 y = 2 N − 1 2 x ≤ y ≤ 4 x 2x\le y\le4x 2 x ≤ y ≤ 4 x O ( N L − D ) O(NL^{-D}) O ( N L − D ) k ≤ x k\le x k ≤ x
Ψ ( 2 N − 1 , 2 k ) = S ( 2 k ) ( 2 N − 1 − 2 k ) + O ( N L D + 1 ) ≥ 2 ω ( 2 a + 5 ) − O ( N L D + 1 ) , \Psi(2N-1,2k)=\mathfrak{S}(2k)(2N-1-2k)+O\left(\frac{N}{L^{D+1}}\right)\ge2\omega(2a+5)-O\left(\frac{N}{L^{D+1}}\right), Ψ ( 2 N − 1 , 2 k ) = S ( 2 k ) ( 2 N − 1 − 2 k ) + O ( L D + 1 N ) ≥ 2 ω ( 2 a + 5 ) − O ( L D + 1 N ) , which exceeds O ( N L 3 ) O(\sqrt{N}L^{3}) O ( N L 3 ) N N N n n n 2 k < n ≤ 2 N − 1 2k<n\le2N-1 2 k < n ≤ 2 N − 1 Q = n Q=n Q = n P = n − 2 k P=n-2k P = n − 2 k N < 2 k < Q < 2 N N<2k<Q<2N N < 2 k < Q < 2 N P ≤ 2 N − 1 − 2 k < N P\le2N-1-2k<N P ≤ 2 N − 1 − 2 k < N P > 2 P>2 P > 2 k ∈ E N k\in E_N k ∈ E N
Conclusion. The remaining integers k ∈ A N k\in A_N k ∈ A N k ≤ a k\le a k ≤ a k > N − 3 − a k>N-3-a k > N − 3 − a 2 a 2a 2 a
∣ A N ∖ E N ∣ ≤ 2 a + O ( N L − D ) ≪ D N / L D . |A_N\setminus E_N|\le2a+O(NL^{-D})\ll_D N/L^D. ∣ A N ∖ E N ∣ ≤ 2 a + O ( N L − D ) ≪ D N / L D . Corollary 3.12 (Goldbach partitions up to a difference of two primes). For every D > 0 D>0 D > 0
∣ C N ∩ E N ∣ = π ( N − 1 ) − 1 + O D ( N ( log N ) D ) . |C_N\cap E_N|=\pi(N-1)-1+O_D\left(\frac{N}{(\log N)^D}\right). ∣ C N ∩ E N ∣ = π ( N − 1 ) − 1 + O D ( ( log N ) D N ) . Equivalently, for all but O D ( N / ( log N ) D ) O_D(N/(\log N)^D) O D ( N / ( log N ) D ) P P P 3 ≤ P < N 3\le P<N 3 ≤ P < N P ′ P' P ′ Q ′ Q' Q ′ 2 < P ′ < N < Q ′ < 2 N 2<P'<N<Q'<2N 2 < P ′ < N < Q ′ < 2 N
Q ′ = ( 2 N − P ) + ( P ′ − P ) . Q'=(2N-P)+(P'-P). Q ′ = ( 2 N − P ) + ( P ′ − P ) . In particular, C N ∩ E N ≠ ∅ C_N\cap E_N\ne\varnothing C N ∩ E N = ∅ N N N P ′ = P P'=P P ′ = P
Proof. The map M ↦ N − M M\mapsto N-M M ↦ N − M C N C_N C N P P P 3 ≤ P ≤ N − 1 3\le P\le N-1 3 ≤ P ≤ N − 1 ∣ C N ∣ = π ( N − 1 ) − 1 |C_N|=\pi(N-1)-1 ∣ C N ∣ = π ( N − 1 ) − 1 ∣ C N ∣ − ∣ A N ∖ E N ∣ ≤ ∣ C N ∩ E N ∣ ≤ ∣ C N ∣ |C_N|-|A_N\setminus E_N|\le|C_N\cap E_N|\le|C_N| ∣ C N ∣ − ∣ A N ∖ E N ∣ ≤ ∣ C N ∩ E N ∣ ≤ ∣ C N ∣ Proposition 3.11 gives the first statement. For a prime 3 ≤ P < N 3\le P<N 3 ≤ P < N M = N − P M=N-P M = N − P A N A_N A N M ∈ E N M\in E_N M ∈ E N 2 ( N − P ) = Q ′ − P ′ 2(N-P)=Q'-P' 2 ( N − P ) = Q ′ − P ′ 2 < P ′ < N < Q ′ < 2 N 2<P'<N<Q'<2N 2 < P ′ < N < Q ′ < 2 N Q ′ = ( 2 N − P ) + ( P ′ − P ) Q'=(2N-P)+(P'-P) Q ′ = ( 2 N − P ) + ( P ′ − P ) π ( N − 1 ) − 1 > ( N − 1 ) / log N − 1 \pi(N-1)-1>(N-1)/\log N-1 π ( N − 1 ) − 1 > ( N − 1 ) / log N − 1 Proposition 2.4 , which exceeds the error term for large N N N
Proposition 3.11 is deduced from Proposition 3.10 and gives no explicit value of N N N Theorem 3.9 gives the explicit bound ∣ E N ∣ > N / 760 |E_N|>N/760 ∣ E N ∣ > N /760 N ≥ 3275 N\ge3275 N ≥ 3275
A Positive Proportion of Centered Partitions
The same energy method applies to the centered count
R ( 2 N ) = ∣ C N ∩ D N ∣ = # { p ∣ 3 ≤ p < N , p and 2 N − p prime } . R(2N)=|C_N\cap D_N|=\#\{p\mid3\le p<N,\ p\text{ and }2N-p\text{ prime}\}. R ( 2 N ) = ∣ C N ∩ D N ∣ = # { p ∣ 3 ≤ p < N , p and 2 N − p prime } . Theorem 3.13 (Positive proportion). For every real number X ≥ 10 6 X\ge10^{6} X ≥ 1 0 6
# { N ∈ Z ∣ 4 ≤ N ≤ X , R ( 2 N ) ≥ 1 } > X 9600 . \#\{N\in\mathbb{Z}\mid4\le N\le X,\ R(2N)\ge1\}>\frac{X}{9600}. # { N ∈ Z ∣ 4 ≤ N ≤ X , R ( 2 N ) ≥ 1 } > 9600 X . Proof. Write L = log X L=\log X L = log X s = X / L s=X/L s = X / L S S S 2 X 2X 2 X
T = ∑ 4 ≤ N ≤ X R ( 2 N ) , V = ∑ 4 ≤ N ≤ X R ( 2 N ) 2 . T=\sum_{4\le N\le X} R(2N),\qquad V=\sum_{4\le N\le X} R(2N)^2. T = 4 ≤ N ≤ X ∑ R ( 2 N ) , V = 4 ≤ N ≤ X ∑ R ( 2 N ) 2 . By the Cauchy–Schwarz inequality, the number of N N N R ( 2 N ) ≥ 1 R(2N)\ge1 R ( 2 N ) ≥ 1 T 2 / V T^2/V T 2 / V
Lower bound for T T T Every pair of odd primes p < q ≤ X p<q\le X p < q ≤ X N = ( p + q ) / 2 N=(p+q)/2 N = ( p + q ) /2 4 ≤ N < X 4\le N<X 4 ≤ N < X M = N − p ∈ C N ∩ D N M=N-p\in C_N\cap D_N M = N − p ∈ C N ∩ D N ( N , p ) (N,p) ( N , p ) R ( 2 N ) R(2N) R ( 2 N ) Proposition 2.4 ,
T ≥ ( π ( X ) − 1 2 ) > ( s − 2 ) 2 2 . T\ge\binom{\pi(X)-1}{2}>\frac{(s-2)^2}{2}. T ≥ ( 2 π ( X ) − 1 ) > 2 ( s − 2 ) 2 . Upper bound for V V V The square R ( 2 N ) 2 R(2N)^2 R ( 2 N ) 2 2 N = p + q = p ′ + q ′ 2N=p+q=p'+q' 2 N = p + q = p ′ + q ′ p < q p<q p < q p ′ < q ′ p'<q' p ′ < q ′ V V V ( p , q , p ′ , q ′ ) ∈ S 4 (p,q,p',q')\in S^4 ( p , q , p ′ , q ′ ) ∈ S 4 p + q = p ′ + q ′ p+q=p'+q' p + q = p ′ + q ′ p ′ − p = q − q ′ = : h p'-p=q-q'=:h p ′ − p = q − q ′ =: h ρ S ( h ) = # { ( p , p ′ ) ∈ S 2 ∣ p ′ − p = h } \rho_S(h)=\#\{(p,p')\in S^2\mid p'-p=h\} ρ S ( h ) = # {( p , p ′ ) ∈ S 2 ∣ p ′ − p = h } ∑ h ρ S ( h ) 2 \sum_h\rho_S(h)^2 ∑ h ρ S ( h ) 2 h = 0 h=0 h = 0 ∣ S ∣ 2 ≤ π ( 2 X ) 2 |S|^2\le\pi(2X)^2 ∣ S ∣ 2 ≤ π ( 2 X ) 2 h ≠ 0 h\ne0 h = 0 ∣ h ∣ ≤ 2 X |h|\le2X ∣ h ∣ ≤ 2 X Proposition 3.6 with x = 2 X x=2X x = 2 X
ρ S ( h ) ≤ 16 ω κ ( h ) 2 X log 2 ( 2 X ) ≤ 32 ω κ ( h ) X L 2 . \rho_S(h)\le16\omega\kappa(h)\frac{2X}{\log^2(2X)}\le32\omega\kappa(h)\frac{X}{L^2}. ρ S ( h ) ≤ 16 ω κ ( h ) log 2 ( 2 X ) 2 X ≤ 32 ω κ ( h ) L 2 X . and ρ S ( h ) = 0 \rho_S(h)=0 ρ S ( h ) = 0 h h h ∣ h ∣ > 2 X |h|>2X ∣ h ∣ > 2 X Lemma 3.8 , ∑ 2 ≤ h ≤ 2 X , h e v e n κ ( h ) 2 ≤ X G \sum_{2\le h\le2X,\ h\ {\rm even}}\kappa(h)^2\le XG ∑ 2 ≤ h ≤ 2 X , h even κ ( h ) 2 ≤ X G
V ≤ π ( 2 X ) 2 + 2 ⋅ 1024 ω 2 X 2 L 4 X G = π ( 2 X ) 2 + K X 3 L 4 , K = 2048 ω 2 G . V\le\pi(2X)^2+2\cdot1024\omega^2\frac{X^2}{L^4}XG=\pi(2X)^2+K\frac{X^3}{L^4},\qquad K=2048\omega^2G. V ≤ π ( 2 X ) 2 + 2 ⋅ 1024 ω 2 L 4 X 2 X G = π ( 2 X ) 2 + K L 4 X 3 , K = 2048 ω 2 G . By Proposition 2.3 (2), and since y ↦ y − 1 ( 1 + 1.2762 / y ) y\mapsto y^{-1}(1+1.2762/y) y ↦ y − 1 ( 1 + 1.2762/ y ) π ( 2 X ) ≤ ( 2 X / L ) ( 1 + 1.2762 / L ) \pi(2X)\le(2X/L)(1+1.2762/L) π ( 2 X ) ≤ ( 2 X / L ) ( 1 + 1.2762/ L )
Conclusion. Since X 3 / L 4 = s 4 / X X^3/L^4=s^4/X X 3 / L 4 = s 4 / X
T 2 V > ( s − 2 ) 4 / 4 4 s 2 ( 1 + 1.2762 / L ) 2 + K s 4 / X = X ( 1 − 2 / s ) 4 4 ( K + e ( X ) ) , \frac{T^2}{V}>\frac{(s-2)^4/4}{4s^2(1+1.2762/L)^2+Ks^4/X}=\frac{X(1-2/s)^4}{4(K+e(X))}, V T 2 > 4 s 2 ( 1 + 1.2762/ L ) 2 + K s 4 / X ( s − 2 ) 4 /4 = 4 ( K + e ( X )) X ( 1 − 2/ s ) 4 , where e ( X ) = 4 L 2 / X ( 1 + 1.2762 / L ) 2 e(X)=4L^2/X(1+1.2762/L)^2 e ( X ) = 4 L 2 / X ( 1 + 1.2762/ L ) 2 X ≥ 10 6 X\ge10^6 X ≥ 1 0 6 s s s e ( X ) e(X) e ( X ) ( 1 − 2 / s ) 4 > 0.99988 (1-2/s)^4>0.99988 ( 1 − 2/ s ) 4 > 0.99988 e ( X ) < 0.00092 e(X)<0.00092 e ( X ) < 0.00092 Lemma 3.8 , K < 2048 ⋅ 0.6614 2 ⋅ 2.658 < 2381.3 K<2048\cdot0.6614^2\cdot2.658<2381.3 K < 2048 ⋅ 0.661 4 2 ⋅ 2.658 < 2381.3
T 2 V > 0.99988 4 ⋅ 2381.30092 X > X 9600 . \frac{T^2}{V}>\frac{0.99988}{4\cdot2381.30092}X>\frac{X}{9600}. V T 2 > 4 ⋅ 2381.30092 0.99988 X > 9600 X . □
Montgomery and Vaughan [13 ] proved that, for some δ > 0 \delta>0 δ > 0 X X X X 1 − δ X^{1-\delta} X 1 − δ n ≤ X n\le X n ≤ X Theorem 3.13 is weaker in its conclusion, but it is explicit and holds for every X ≥ 10 6 X\ge10^6 X ≥ 1 0 6 Proposition 5.7 , its conclusion is trivial for X ≤ 2 ⋅ 10 18 X\le2\cdot10^{18} X ≤ 2 ⋅ 1 0 18
A Half-Difference Coverage Statistic
For the diagnostic set E N E_N E N
H ( N ) = log 2 ( 2 N ) − ( ( N − 3 ) − ∣ E N ∣ ) . H(N)=\log^2(2N)-((N-3)-|E_N|). H ( N ) = log 2 ( 2 N ) − (( N − 3 ) − ∣ E N ∣ ) . Rearranging, H ( N ) > 0 H(N)>0 H ( N ) > 0
∣ E N ∣ > ( N − 3 ) − log 2 ( 2 N ) . |E_N|>(N-3)-\log^2(2N). ∣ E N ∣ > ( N − 3 ) − log 2 ( 2 N ) . Thus H ( N ) > 0 H(N)>0 H ( N ) > 0 log 2 ( 2 N ) \log^2(2N) log 2 ( 2 N ) E N E_N E N
Hypothesis 3.14 (Half-Difference Density Hypothesis). For every integer N > 10 6 N>10^6 N > 1 0 6 H ( N ) > 0 H(N)>0 H ( N ) > 0
Hypothesis 3.15 (Centered Prime-Correlation Hypothesis). For every integer N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18 the centered prime-pair count
R ( 2 N ) : = ∣ C N ∩ D N ∣ R(2N) := |C_N \cap D_N| R ( 2 N ) := ∣ C N ∩ D N ∣ is positive.
Computational Evidence
The computations use two scripts of the repository [14 ] : experiment.py, which uses only the Python 3.12 standard library, and large_range.py, which also uses NumPy. The scripts were written with Claude (Claude Opus 5.5) and run by the author; the running times below are machine dependent.
The script experiment.py computes ∣ E N ∣ |E_N| ∣ E N ∣ H ( N ) H(N) H ( N ) ∣ C N ∣ |C_N| ∣ C N ∣ R ( 2 N ) = ∣ C N ∩ D N ∣ R(2N) = |C_N \cap D_N| R ( 2 N ) = ∣ C N ∩ D N ∣ ( C N ∩ E N ) ∖ D N (C_N \cap E_N) \setminus D_N ( C N ∩ E N ) ∖ D N N ∈ [ 4 , 2 14 ] N \in[4, 2^{14}] N ∈ [ 4 , 2 14 ] 2 15 2^{15} 2 15 E N E_N E N large_range.py computes ∣ E N ∣ |E_N| ∣ E N ∣ R ( 2 N ) R(2N) R ( 2 N ) N ∈ [ 4 , 10 6 ] N \in[4, 10^6] N ∈ [ 4 , 1 0 6 ] ( P , Q ) (P,Q) ( P , Q ) Q − P Q-P Q − P N N N R ( 2 N ) R(2N) R ( 2 N ) [ 4 , 2 14 ] [4, 2^{14}] [ 4 , 2 14 ] H ( N ) H(N) H ( N )
I 2 = [ 4 , 8 ] , I m = ( 2 m , 2 m + 1 ] ( m ≥ 3 ) . I_2 = [4,8], \qquad I_m = (2^m,2^{m+1}] \quad(m \ge3). I 2 = [ 4 , 8 ] , I m = ( 2 m , 2 m + 1 ] ( m ≥ 3 ) . Table 1 gives, for 2 ≤ m ≤ 19 2 \le m \le19 2 ≤ m ≤ 19 H ( N ) H(N) H ( N ) I m I_m I m A N ∖ E N A_N \setminus E_N A N ∖ E N I m I_m I m ∣ E N ∣ / N |E_N|/N ∣ E N ∣/ N I m I_m I m H ( N ) > 0 H(N) > 0 H ( N ) > 0 N ∈ [ 4 , 10 6 ] N \in[4,10^6] N ∈ [ 4 , 1 0 6 ] Hypothesis 3.14 . Second, the minimum of H ( N ) H(N) H ( N ) I m I_m I m 2 ≤ m ≤ 16 2 \le m \le16 2 ≤ m ≤ 16 I 17 I_{17} I 17 I 16 I_{16} I 16 I 17 I_{17} I 17 I 19 I_{19} I 19 I 19 I_{19} I 19 N = 668364 N = 668364 N = 668364 log 2 ( 2 N ) > 198 \log^2(2N) > 198 log 2 ( 2 N ) > 198
Table 1. Minimum H ( N ) H(N) H ( N ) A N ∖ E N A_N \setminus E_N A N ∖ E N N N N ∣ E N ∣ / N |E_N|/N ∣ E N ∣/ N I m I_m I m H ( N ) > 0 H(N) > 0 H ( N ) > 0
Centered Partitions and the Pigeonhole Count
The run of experiment.py gives the following results for every N ∈ [ 4 , 2 14 ] N \in[4,2^{14}] N ∈ [ 4 , 2 14 ]
R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 2 N 2N 2 N 8 ≤ 2 N ≤ 2 15 8 \le2N \le2^{15} 8 ≤ 2 N ≤ 2 15 Lemma 3.3 , C N ∩ E N ⊇ C N ∩ D N ≠ ∅ C_N \cap E_N \supseteq C_N \cap D_N \ne\varnothing C N ∩ E N ⊇ C N ∩ D N = ∅
The pigeonhole count ∣ C N ∣ > ∣ A N ∖ E N ∣ |C_N| > |A_N \setminus E_N| ∣ C N ∣ > ∣ A N ∖ E N ∣ N N N N = 5 N = 5 N = 5 N = 7 N = 7 N = 7 C 5 ∩ E 5 = { 2 } C_5 \cap E_5 = \{2\} C 5 ∩ E 5 = { 2 } C 7 ∩ E 7 = { 4 } C_7 \cap E_7 = \{4\} C 7 ∩ E 7 = { 4 }
( C N ∩ E N ) ∖ D N ≠ ∅ (C_N \cap E_N) \setminus D_N \ne\varnothing ( C N ∩ E N ) ∖ D N = ∅ N N N N N N N ∈ { 4 , 5 , 7 , 8 } N \in\{4,5,7,8\} N ∈ { 4 , 5 , 7 , 8 } M = 3 M = 3 M = 3 N = 6 N = 6 N = 6 M = 6 M = 6 M = 6 N = 9 N = 9 N = 9 M = 5 M = 5 M = 5 N = 10 N = 10 N = 10 M = 4 M = 4 M = 4 N = 11 N = 11 N = 11 M = 9 M = 9 M = 9 N = 12 N = 12 N = 12 3 ∈ C 6 ∩ E 6 3 \in C_6 \cap E_6 3 ∈ C 6 ∩ E 6 6 + 3 = 9 6 + 3 = 9 6 + 3 = 9
Among the integers 2 ≤ N ≤ 10 6 2 \le N \le10^6 2 ≤ N ≤ 1 0 6 Lemma 2.6 fails at N = 355 N = 355 N = 355 N ≥ 356 N \ge356 N ≥ 356 Lemma 2.6 is therefore the least possible integer threshold.
For comparison with H ( N ) H(N) H ( N ) H R ( N ) H_R(N) H R ( N ) § 5.3 on the intervals I m I_m I m Table 2 . The minimum is positive on I 2 I_2 I 2 I 3 I_3 I 3 I m I_m I m 4 ≤ m ≤ 13 4 \le m \le13 4 ≤ m ≤ 13 Proposition 5.6 , H R ( N ) < 0 H_R(N) < 0 H R ( N ) < 0 N ≥ 34 N \ge34 N ≥ 34
Table 2. Minimum H R ( N ) H_R(N) H R ( N ) I m I_m I m
The Exceptional Set and the Linear Bound
The minimum of ∣ E N ∣ / N |E_N|/N ∣ E N ∣/ N Theorem 3.9 gives ∣ E N ∣ / N > 1 / 760 |E_N|/N > 1/760 ∣ E N ∣/ N > 1/760 N ≥ 3275 N \ge3275 N ≥ 3275 Proposition 3.11 gives ∣ E N ∣ / N → 1 |E_N|/N \to1 ∣ E N ∣/ N → 1 N N N Table 1 at which the minimum of H ( N ) H(N) H ( N ) A N ∖ E N A_N \setminus E_N A N ∖ E N N ≤ 2161 N \le2161 N ≤ 2161 M ≤ 50 M \le50 M ≤ 50 N − 3 − M ≤ 14 N - 3 - M \le14 N − 3 − M ≤ 14 N = 1327 N = 1327 N = 1327 M = 1 , … , 19 , 22 , 25 , 28 , 34 , 43 , 50 , 1316 , 1319 , 1323 M = 1,\ldots,19,22,25,28,34,43,50,1316,1319,1323 M = 1 , … , 19 , 22 , 25 , 28 , 34 , 43 , 50 , 1316 , 1319 , 1323 p − = 1321 p^- = 1321 p − = 1321 p + = 1361 p^+ = 1361 p + = 1361 Lemma 3.4 forces exactly the missing values M ≤ 19 M \le19 M ≤ 19 ( p + − p − ) / 2 = 20 (p^+ - p^-)/2 = 20 ( p + − p − ) /2 = 20 p + − p − = 26 p^+ - p^- = 26 p + − p − = 26 N = 2161 N = 2161 N = 2161 p + − p − = 24 p^+ - p^- = 24 p + − p − = 24 N = 541 N = 541 N = 541 M = 1 , … , 12 M = 1,\ldots,12 M = 1 , … , 12 M = 1 , … , 11 M = 1,\ldots,11 M = 1 , … , 11 N N N Lemma 3.5 forces the final runs. The largest primes below 2 N 2N 2 N N = 269 N = 269 N = 269 q = 523 q = 523 q = 523 M = 261 , … , 266 M = 261,\ldots,266 M = 261 , … , 266 M = 534 , … , 538 M = 534,\ldots,538 M = 534 , … , 538 M = 2148 , … , 2158 M = 2148,\ldots,2158 M = 2148 , … , 2158 N = 269 N = 269 N = 269 N − 3 N - 3 N − 3 N = 2161 N = 2161 N = 2161 N − 3 − M ≤ 14 N - 3 - M \le14 N − 3 − M ≤ 14 M = 2144 M = 2144 M = 2144
Comparison with the Hardy–Littlewood Prediction
Let
HL ( 2 N ) = ω ∏ p ∣ N p > 2 p − 1 p − 2 ∫ 2 2 N − 2 d t log t log ( 2 N − t ) \operatorname{HL}(2N) = \omega\prod_{\substack{p \mid N \\ p > 2}} \frac{p-1}{p-2} \int_{2}^{2N-2} \frac{\mathrm{d}t}{\log t \log(2N-t)} HL ( 2 N ) = ω p ∣ N p > 2 ∏ p − 2 p − 1 ∫ 2 2 N − 2 log t log ( 2 N − t ) d t with ω \omega ω Proposition 3.6 . The script large_range.py computes the integral by 64-point Gauss–Legendre quadrature and ω \omega ω 2 ⋅ 10 6 2 \cdot10^{6} 2 ⋅ 1 0 6 ω ≈ 0.6601618372 \omega\approx0.6601618372 ω ≈ 0.6601618372 Table 3 gives, on each interval I m I_m I m R ( 2 N ) R(2N) R ( 2 N ) R ( 2 N ) / HL ( 2 N ) R(2N)/\operatorname{HL}(2N) R ( 2 N ) / HL ( 2 N ) R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 N ∈ [ 4 , 10 6 ] N \in[4,10^{6}] N ∈ [ 4 , 1 0 6 ] 2 N 2N 2 N 8 ≤ 2 N ≤ 2 ⋅ 10 6 8 \le2N \le2 \cdot10^{6} 8 ≤ 2 N ≤ 2 ⋅ 1 0 6 I 19 I_{19} I 19
Table 3. Minimum of R ( 2 N ) R(2N) R ( 2 N ) N N N R ( 2 N ) / HL ( 2 N ) R(2N)/\operatorname{HL}(2N) R ( 2 N ) / HL ( 2 N ) I m I_m I m
Missing Values Forced by Prime Gaps
For each N N N F ( N ) F(N) F ( N ) M ∈ A N M \in A_N M ∈ A N Lemma 3.4 or Lemma 3.5 places in A N ∖ E N A_N \setminus E_N A N ∖ E N U ( N ) = ∣ A N ∖ E N ∣ − F ( N ) U(N) = |A_N \setminus E_N| - F(N) U ( N ) = ∣ A N ∖ E N ∣ − F ( N ) large_range.py computes F ( N ) F(N) F ( N ) U ( N ) U(N) U ( N ) N ∈ [ 4 , 10 6 ] N \in[4,10^{6}] N ∈ [ 4 , 1 0 6 ] N N N 2 N 2N 2 N F ( N ) ≤ ∣ A N ∖ E N ∣ F(N) \le|A_N \setminus E_N| F ( N ) ≤ ∣ A N ∖ E N ∣ N N N Table 4 gives, on each interval I m I_m I m ∣ A N ∖ E N ∣ |A_N \setminus E_N| ∣ A N ∖ E N ∣ F ( N ) F(N) F ( N ) U ( N ) U(N) U ( N ) U ( N ) U(N) U ( N ) ∑ F ( N ) / ∑ ∣ A N ∖ E N ∣ \sum F(N)/\sum|A_N \setminus E_N| ∑ F ( N ) / ∑ ∣ A N ∖ E N ∣
Table 4. Largest exceptional set, largest number F ( N ) F(N) F ( N ) U ( N ) U(N) U ( N ) N N N U ( N ) U(N) U ( N ) I m I_m I m
The share of forced values strictly decreases with m m m I 2 I_2 I 2 I 19 I_{19} I 19 U ( N ) U(N) U ( N ) I 6 I_6 I 6 I m I_m I m 8 ≤ m ≤ 14 8 \le m \le14 8 ≤ m ≤ 14 N N N I m I_m I m 15 ≤ m ≤ 19 15 \le m \le19 15 ≤ m ≤ 19 N N N U ( N ) = 0 U(N) = 0 U ( N ) = 0 N ≤ 10 6 N \le10^{6} N ≤ 1 0 6 F ( N ) ≥ log 2 ( 2 N ) F(N) \ge\log^{2}(2N) F ( N ) ≥ log 2 ( 2 N ) U ( N ) > log 2 ( 2 N ) U(N) > \log^{2}(2N) U ( N ) > log 2 ( 2 N ) U ( N ) / log 2 ( 2 N ) U(N)/\log^{2}(2N) U ( N ) / log 2 ( 2 N ) 1000 ≤ N ≤ 10 6 1000 \le N \le10^{6} 1000 ≤ N ≤ 1 0 6 N = 484857 N = 484857 N = 484857
Rigorous Consequences and the Open Centered Problem
What Follows from the Short-Interval Estimate
The results of Dusart imply useful lower bounds for primes above N N N E N E_N E N
Proposition 5.1 (Primes in ( N , 2 N ) (N,2N) ( N , 2 N ) For every integer N ≥ 3275 N \ge3275 N ≥ 3275 the interval ( N , 2 N ) (N,2N) ( N , 2 N ) contains at least ⌊ log 2 N ⌋ − 1 \lfloor\log^2 N\rfloor- 1 ⌊ log 2 N ⌋ − 1 primes.
Proof. Starting with x 0 = N x_0=N x 0 = N
( x j , x j ( 1 + 1 / ( 2 log 2 x j ) ) ] , j = 0 , 1 , 2 , … , (x_j, x_j(1 + 1/(2\log^2 x_j))], \qquad j=0,1,2,\ldots, ( x j , x j ( 1 + 1/ ( 2 log 2 x j ))] , j = 0 , 1 , 2 , … , until the first index m + 1 m+1 m + 1 x m + 1 ≥ 2 N x_{m+1} \ge2N x m + 1 ≥ 2 N Proposition 2.1 , each of the first m m m ( N , 2 N ) (N,2N) ( N , 2 N ) x j ≤ 2 N x_j \le2N x j ≤ 2 N log x j ≥ log N \log x_j \ge\log N log x j ≥ log N N / log 2 N N/\log^2 N N / log 2 N N N N log 2 N \log^2 N log 2 N m ≥ log 2 N − 1 m \ge\log^2 N - 1 m ≥ log 2 N − 1 ( N , 2 N ) (N,2N) ( N , 2 N ) ⌊ log 2 N ⌋ − 1 \lfloor\log^2 N\rfloor- 1 ⌊ log 2 N ⌋ − 1
Corollary 5.2 (Baseline half-difference coverage). For every integer N ≥ 3275 N \ge3275 N ≥ 3275
∣ E N ∣ ≥ ⌊ log 2 N ⌋ − 2. |E_N| \ge\lfloor\log^2 N\rfloor- 2. ∣ E N ∣ ≥ ⌊ log 2 N ⌋ − 2. Proof. Let Q Q Q ( N , 2 N ) (N,2N) ( N , 2 N ) Q ≤ 2 N − 3 Q \le2N - 3 Q ≤ 2 N − 3 P = 3 P=3 P = 3 M = ( Q − 3 ) / 2 M=(Q-3)/2 M = ( Q − 3 ) /2 A N A_N A N E N E_N E N Q Q Q M M M ( N , 2 N ) (N,2N) ( N , 2 N ) Q = 2 N − 1 Q=2N-1 Q = 2 N − 1 M = N − 2 M=N-2 M = N − 2 Proposition 5.1 therefore gives the stated lower bound.
Theorem 3.9 gives the linear bound ∣ E N ∣ > N / 760 |E_N| > N/760 ∣ E N ∣ > N /760 N N N Hypothesis 3.14 asks for ∣ E N ∣ > ( N − 3 ) − log 2 ( 2 N ) |E_N| > (N-3)-\log^2(2N) ∣ E N ∣ > ( N − 3 ) − log 2 ( 2 N ) R ( 2 N ) = ∣ C N ∩ D N ∣ > 0 R(2N)=|C_N \cap D_N| > 0 R ( 2 N ) = ∣ C N ∩ D N ∣ > 0
Proposition 5.3 (A pigeonhole consequence of H ( N ) > 0 H(N)>0 H ( N ) > 0 For N ≥ 356 N \ge356 N ≥ 356 if H ( N ) > 0 H(N)>0 H ( N ) > 0 then
C N ∩ E N ≠ ∅ . C_N \cap E_N \ne\varnothing. C N ∩ E N = ∅ . Proof. The condition H ( N ) > 0 H(N) > 0 H ( N ) > 0
∣ A N ∖ E N ∣ < log 2 ( 2 N ) . \lvert A_N \setminus E_N\rvert< \log^2(2N). ∣ A N ∖ E N ∣ < log 2 ( 2 N ) . On the other hand, ∣ C N ∣ = π ( N − 1 ) − 1 ≥ π ( N ) − 2 \lvert C_N\rvert= \pi(N - 1) - 1 \ge\pi(N) - 2 ∣ C N ∣ = π ( N − 1 ) − 1 ≥ π ( N ) − 2 Proposition 2.4 , since N ≥ 356 > 17 N \ge356 > 17 N ≥ 356 > 17
π ( N ) > N log N > N log N + 2 . \pi(N) > \frac{N}{\log N} > \frac{N}{\log N + 2}. π ( N ) > log N N > log N + 2 N . and by Lemma 2.6 ,
∣ C N ∣ > N log N + 2 − 2 > log 2 ( 2 N ) . \lvert C_N\rvert> \frac{N}{\log N + 2} - 2 > \log^2(2N). ∣ C N ∣ > log N + 2 N − 2 > log 2 ( 2 N ) . Hence C N C_N C N A N ∖ E N A_N \setminus E_N A N ∖ E N C N ∩ E N C_N \cap E_N C N ∩ E N
Proposition 5.3 gives C N ∩ E N ≠ ∅ C_N \cap E_N \ne\varnothing C N ∩ E N = ∅ N N N H ( N ) > 0 H(N) > 0 H ( N ) > 0 § 4 verifies for 4 ≤ N ≤ 10 6 4 \le N \le10^6 4 ≤ N ≤ 1 0 6 N N N N − M N - M N − M N N N M M M 2 N 2N 2 N N − M N - M N − M ( N − M , N + M ) (N - M, N + M) ( N − M , N + M )
Prime Gaps and the Coverage Statistic
Lemma 3.4 turns every long prime gap into many missing values of E N E_N E N H ( N ) H(N) H ( N )
Theorem 5.4 (Coverage implies a Cramér-type bound). Let N 0 ≥ 4 N_0 \ge4 N 0 ≥ 4 and suppose that H ( N ) > 0 H(N) > 0 H ( N ) > 0 for every integer N ≥ N 0 N \ge N_0 N ≥ N 0 Then every pair of consecutive primes p < p ′ p < p' p < p ′ with $p \ge \max{3, N_0 - 1} satisfies
p ′ − p < 2 log 2 ( 2 p + 2 ) + 2. p' - p < 2\log^2(2p + 2) + 2. p ′ − p < 2 log 2 ( 2 p + 2 ) + 2. Proof. Let N = p + 1 N = p + 1 N = p + 1 N ≥ N 0 N \ge N_0 N ≥ N 0 N ≥ 4 N \ge4 N ≥ 4 N N N p p p N N N p ′ p' p ′ p ′ ≥ p + 2 > N p' \ge p + 2 > N p ′ ≥ p + 2 > N g = p ′ − p g = p' - p g = p ′ − p Lemma 3.4 , the values M = 1 , … , g / 2 − 1 M = 1, \ldots, g/2 - 1 M = 1 , … , g /2 − 1 A N ∖ E N A_N \setminus E_N A N ∖ E N A N A_N A N p ′ < 2 p p' < 2p p ′ < 2 p g < p = N − 1 g < p = N - 1 g < p = N − 1 g / 2 − 1 ≤ N − 3 g/2 - 1 \le N - 3 g /2 − 1 ≤ N − 3 ∣ A N ∖ E N ∣ ≥ g / 2 − 1 \lvert A_N \setminus E_N\rvert\ge g/2 - 1 ∣ A N ∖ E N ∣ ≥ g /2 − 1 H ( N ) > 0 H(N) > 0 H ( N ) > 0 ∣ A N ∖ E N ∣ < log 2 ( 2 N ) \lvert A_N \setminus E_N\rvert< \log^2(2N) ∣ A N ∖ E N ∣ < log 2 ( 2 N ) g / 2 − 1 < log 2 ( 2 p + 2 ) g/2 - 1 < \log^2(2p + 2) g /2 − 1 < log 2 ( 2 p + 2 )
Cramér [15 ] conjectured that p ′ − p = O ( log 2 p ) p' - p = O(\log^2 p) p ′ − p = O ( log 2 p ) p < p ′ p < p' p < p ′ p ′ − p ≤ p 0.525 p' - p \le p^{0.525} p ′ − p ≤ p 0.525 p p p [16 ] . Since the computation of § 4 gives H ( N ) > 0 H(N) > 0 H ( N ) > 0 4 ≤ N ≤ 10 6 4 \le N \le10^6 4 ≤ N ≤ 1 0 6 Theorem 5.4 shows that Hypothesis 3.14 implies p ′ − p < 2 log 2 ( 2 p + 2 ) + 2 p' - p < 2\log^2(2p + 2) + 2 p ′ − p < 2 log 2 ( 2 p + 2 ) + 2 p < p ′ p < p' p < p ′ p ≥ 3 p \ge3 p ≥ 3
In the other direction, long prime gaps give large exceptional sets. Ford, Green, Konyagin, Maynard and Tao [8 ] proved that there is an absolute constant c 0 > 0 c_0 > 0 c 0 > 0 X X X
max p n ≤ X ( p n + 1 − p n ) ≥ c 0 F ( X ) , F ( X ) = log X log log X log log log log X log log log X , \max_{p_n \le X}(p_{n+1} - p_n) \ge c_0 F(X), \qquad F(X) = \frac{\log X \log\log X \log\log\log\log X}{\log\log\log X}, p n ≤ X max ( p n + 1 − p n ) ≥ c 0 F ( X ) , F ( X ) = log log log X log X log log X log log log log X , where p n p_n p n n n n
Corollary 5.5 (Large exceptional sets). There is an absolute constant c > 0 c > 0 c > 0 such that ∣ A N ∖ E N ∣ ≥ c F ( N ) \lvert A_N \setminus E_N\rvert\ge cF(N) ∣ A N ∖ E N ∣ ≥ c F ( N ) for infinitely many integers N N N In particular, ∣ A N ∖ E N ∣ \lvert A_N \setminus E_N\rvert ∣ A N ∖ E N ∣ is unbounded.
Proof. For large X X X p n < p n + 1 p_n < p_{n+1} p n < p n + 1 p n ≤ X p_n \le X p n ≤ X p n + 1 − p n ≥ c 0 F ( X ) p_{n+1} - p_n \ge c_0F(X) p n + 1 − p n ≥ c 0 F ( X ) N = p n + 1 ≤ X + 1 N = p_n + 1 \le X + 1 N = p n + 1 ≤ X + 1 Theorem 5.4 , ∣ A N ∖ E N ∣ ≥ ( p n + 1 − p n ) / 2 − 1 ≥ c 0 F ( X ) / 2 − 1 \lvert A_N \setminus E_N\rvert\ge(p_{n+1} - p_n)/2 - 1 \ge c_0F(X)/2 - 1 ∣ A N ∖ E N ∣ ≥ ( p n + 1 − p n ) /2 − 1 ≥ c 0 F ( X ) /2 − 1 p n > p n + 1 − p n ≥ c 0 F ( X ) p_n > p_{n+1} - p_n \ge c_0F(X) p n > p n + 1 − p n ≥ c 0 F ( X ) N → ∞ N \to\infty N → ∞ X → ∞ X \to\infty X → ∞ N N N F F F N ≤ X + 1 N \le X + 1 N ≤ X + 1 F ( N ) ≤ F ( X + 1 ) F(N) \le F(X + 1) F ( N ) ≤ F ( X + 1 ) F ( X + 1 ) ∼ F ( X ) F(X + 1) \sim F(X) F ( X + 1 ) ∼ F ( X ) ∣ A N ∖ E N ∣ ≥ c F ( N ) \lvert A_N \setminus E_N\rvert\ge cF(N) ∣ A N ∖ E N ∣ ≥ c F ( N ) c < c 0 / 2 c < c_0/2 c < c 0 /2 X X X
Since F ( N ) = o ( log 2 N ) F(N) = o(\log^2 N) F ( N ) = o ( log 2 N ) Hypothesis 3.14 .
The Centered Statistic
For the centered sets, the analogue of H ( N ) H(N) H ( N ) N ≥ 34 N \ge34 N ≥ 34
H R ( N ) = log 2 ( 2 N ) − ( ( N − 3 ) − R ( 2 N ) ) , H_R(N) = \log^2(2N) - ((N - 3) - R(2N)), H R ( N ) = log 2 ( 2 N ) − (( N − 3 ) − R ( 2 N )) , obtained from H ( N ) H(N) H ( N ) E N E_N E N C N ∩ D N C_N \cap D_N C N ∩ D N
Proposition 5.6 (Negativity of the centered statistic). For every integer N ≥ 34 N \ge34 N ≥ 34 H R ( N ) < 0 H_R(N) < 0 H R ( N ) < 0
Proof. Write L = log N L = \log N L = log N R ( 2 N ) ≤ ∣ C N ∣ = π ( N − 1 ) − 1 < π ( N ) R(2N) \le|C_N| = \pi(N - 1) - 1 < \pi(N) R ( 2 N ) ≤ ∣ C N ∣ = π ( N − 1 ) − 1 < π ( N ) Proposition 2.3 gives H R ( N ) < − f ( N ) H_R(N) < -f(N) H R ( N ) < − f ( N )
f ( x ) = x − 3 − x log x ( 1 + 1.2762 log x ) − log 2 ( 2 x ) . f(x) = x - 3 - \frac{x}{\log x}\left(1 + \frac{1.2762}{\log x}\right) - \log^2(2x). f ( x ) = x − 3 − log x x ( 1 + log x 1.2762 ) − log 2 ( 2 x ) . For x ≥ 8 x \ge8 x ≥ 8 ℓ = log x \ell= \log x ℓ = log x
f ′ ( x ) = 1 − ℓ − 1 ℓ 2 − 1.2762 ℓ − 2 ℓ 3 − 2 log ( 2 x ) x ≥ 1 − 1 4 − 1.2762 27 − 2 log 16 8 > 0 , f'(x) = 1 - \frac{\ell- 1}{\ell^2} - 1.2762\frac{\ell- 2}{\ell^3} - \frac{2\log(2x)}{x} \ge1 - \frac{1}{4} - \frac{1.2762}{27} - \frac{2\log16}{8} > 0, f ′ ( x ) = 1 − ℓ 2 ℓ − 1 − 1.2762 ℓ 3 ℓ − 2 − x 2 log ( 2 x ) ≥ 1 − 4 1 − 27 1.2762 − 8 2 log 16 > 0 , because ( ℓ − 1 ) / ℓ 2 ≤ 1 / 4 (\ell- 1)/\ell^2 \le1/4 ( ℓ − 1 ) / ℓ 2 ≤ 1/4 ( ℓ − 2 ) / ℓ 3 ≤ 1 / 27 (\ell- 2)/\ell^3 \le1/27 ( ℓ − 2 ) / ℓ 3 ≤ 1/27 ℓ > 0 \ell> 0 ℓ > 0 2 log ( 2 x ) / x 2\log(2x)/x 2 log ( 2 x ) / x x ≥ 8 x \ge8 x ≥ 8 f f f [ 8 , ∞ ) [8,\infty) [ 8 , ∞ ) x = 34 x = 34 x = 34
34 log 34 ( 1 + 1.2762 log 34 ) < 13.132 and log 2 68 < 17.805 , \frac{34}{\log34}\left(1 + \frac{1.2762}{\log34}\right) < 13.132 \quad\text{and} \quad\log^2 68 < 17.805, log 34 34 ( 1 + log 34 1.2762 ) < 13.132 and log 2 68 < 17.805 , so f ( 34 ) > 34 − 3 − 13.132 − 17.805 > 0.06 f(34) > 34 - 3 - 13.132 - 17.805 > 0.06 f ( 34 ) > 34 − 3 − 13.132 − 17.805 > 0.06 f ( N ) > 0 f(N) > 0 f ( N ) > 0 H R ( N ) < 0 H_R(N) < 0 H R ( N ) < 0 N ≥ 34 N \ge34 N ≥ 34
The Gap to a Complete Proof
The missing ingredient is a uniform lower bound for the centered binary prime correlation
R ( 2 N ) = ∣ C N ∩ D N ∣ = # { p < N ∣ p and 2 N − p are prime } . R(2N) = |C_N \cap D_N| = \#\{p < N \mid p \text{ and } 2N - p \text{ are prime}\}. R ( 2 N ) = ∣ C N ∩ D N ∣ = # { p < N ∣ p and 2 N − p are prime } . By Proposition 5.7 , a complete proof would follow from
R ( 2 N ) > 0 ( N > 2 ⋅ 10 18 ) . R(2N) > 0 \qquad(N > 2 \cdot10^{18}). R ( 2 N ) > 0 ( N > 2 ⋅ 1 0 18 ) . Equivalently, the set of primes below N N N N N N Lemma 3.3 , for all large N N N C N ∩ E N C_N \cap E_N C N ∩ E N π ( N − 1 ) − 1 − O D ( N / ( log N ) D ) \pi(N - 1) - 1 - O_D(N/(\log N)^D) π ( N − 1 ) − 1 − O D ( N / ( log N ) D ) C N ∩ D N = C N ∩ E N ∩ D N C_N \cap D_N = C_N \cap E_N \cap D_N C N ∩ D N = C N ∩ E N ∩ D N M M M N + M N + M N + M ( N − M , N + M ) (N - M, N + M) ( N − M , N + M ) M M M 2 N 2N 2 N M M M Proposition 5.6 , ∣ C N ∩ D N ∣ < ( N − 3 ) − log 2 ( 2 N ) |C_N \cap D_N| < (N - 3) - \log^2(2N) ∣ C N ∩ D N ∣ < ( N − 3 ) − log 2 ( 2 N ) N ≥ 34 N \ge34 N ≥ 34 ∣ E N ∣ = N − o ( N ) |E_N| = N - o(N) ∣ E N ∣ = N − o ( N ) Proposition 3.11 .
A strong sufficient form would be an explicit Hardy–Littlewood-type lower bound
R ( 2 N ) ≥ c N log 2 N ∏ p ∣ N p > 2 p − 1 p − 2 R(2N) \ge c \frac{N}{\log^2 N}\prod_{p \mid N \atop p > 2}\frac{p - 1}{p - 2} R ( 2 N ) ≥ c log 2 N N p > 2 p ∣ N ∏ p − 2 p − 1 with a positive constant c c c N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18 R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18
The Verified Range
Proposition 5.7 (Verified range). For every integer N N N 4 ≤ N ≤ 2 ⋅ 10 18 4 \le N \le2 \cdot10^{18} 4 ≤ N ≤ 2 ⋅ 1 0 18 R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1
Proof. Oliveira e Silva, Herzog and Pardi verified the even Goldbach conjecture for all even integers up to 4 ⋅ 10 18 4 \cdot10^{18} 4 ⋅ 1 0 18 [2 ] , and the least prime of a Goldbach partition of every such integer is at most 9781 [17 ] . Hence 2 N = p + q 2N = p + q 2 N = p + q p ≤ 9781 p \le9781 p ≤ 9781 q = 2 N − p q = 2N - p q = 2 N − p N > 9781 N > 9781 N > 9781 p < N < q p < N < q p < N < q p ≥ 3 p \ge3 p ≥ 3 q q q M = N − p ∈ C N ∩ D N M = N - p \in C_N \cap D_N M = N − p ∈ C N ∩ D N R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 4 ≤ N ≤ 9781 4 \le N \le9781 4 ≤ N ≤ 9781 § 4 gives R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1
Consequently, the distinct-prime Goldbach conjecture is equivalent to R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18
Small Cases
The first cases illustrate the centered sets explicitly:
N = 4 N = 4 N = 4 2 N = 8 2N = 8 2 N = 8 C 4 = { 1 } C_4 = \{1\} C 4 = { 1 } D 4 = { 1 } D_4 = \{1\} D 4 = { 1 } 8 = 3 + 5 8 = 3 + 5 8 = 3 + 5
N = 5 N = 5 N = 5 2 N = 10 2N = 10 2 N = 10 C 5 = { 2 } C_5 = \{2\} C 5 = { 2 } D 5 = { 2 } D_5 = \{2\} D 5 = { 2 } 10 = 3 + 7 10 = 3 + 7 10 = 3 + 7
N = 6 N = 6 N = 6 2 N = 12 2N = 12 2 N = 12 C 6 = { 3 , 1 } C_6 = \{3, 1\} C 6 = { 3 , 1 } D 6 = { 1 } D_6 = \{1\} D 6 = { 1 } 12 = 5 + 7 12 = 5 + 7 12 = 5 + 7
N = 7 N = 7 N = 7 2 N = 14 2N = 14 2 N = 14 C 7 = { 4 , 2 } C_7 = \{4, 2\} C 7 = { 4 , 2 } D 7 = { 4 } D_7 = \{4\} D 7 = { 4 } 14 = 3 + 11 14 = 3 + 11 14 = 3 + 11
N = 8 N = 8 N = 8 2 N = 16 2N = 16 2 N = 16 C 8 = { 5 , 3 , 1 } C_8 = \{5, 3, 1\} C 8 = { 5 , 3 , 1 } D 8 = { 3 , 5 } D_8 = \{3, 5\} D 8 = { 3 , 5 } 16 = 3 + 13 16 = 3 + 13 16 = 3 + 13
N = 9 N = 9 N = 9 2 N = 18 2N = 18 2 N = 18 C 9 = { 6 , 4 , 2 } C_9 = \{6, 4, 2\} C 9 = { 6 , 4 , 2 } D 9 = { 2 , 4 } D_9 = \{2, 4\} D 9 = { 2 , 4 } 18 = 5 + 13 18 = 5 + 13 18 = 5 + 13
N = 10 N = 10 N = 10 2 N = 20 2N = 20 2 N = 20 C 10 = { 7 , 5 , 3 } C_{10} = \{7, 5, 3\} C 10 = { 7 , 5 , 3 } D 10 = { 1 , 3 , 7 } D_{10} = \{1, 3, 7\} D 10 = { 1 , 3 , 7 } 20 = 3 + 17 20 = 3 + 17 20 = 3 + 17
N = 11 N = 11 N = 11 2 N = 22 2N = 22 2 N = 22 C 11 = { 8 , 6 , 4 } C_{11} = \{8, 6, 4\} C 11 = { 8 , 6 , 4 } D 11 = { 2 , 6 , 8 } D_{11} = \{2, 6, 8\} D 11 = { 2 , 6 , 8 } 22 = 3 + 19 22 = 3 + 19 22 = 3 + 19
N = 12 N = 12 N = 12 2 N = 24 2N = 24 2 N = 24 C 12 = { 9 , 7 , 5 , 1 } C_{12} = \{9, 7, 5, 1\} C 12 = { 9 , 7 , 5 , 1 } D 12 = { 1 , 5 , 7 } D_{12} = \{1, 5, 7\} D 12 = { 1 , 5 , 7 } 24 = 5 + 19 24 = 5 + 19 24 = 5 + 19
For 13 ≤ N ≤ 2 14 13 \le N \le2^{14} 13 ≤ N ≤ 2 14 § 4 verifies at least one centered intersection value M ∈ C N ∩ D N M \in C_N \cap D_N M ∈ C N ∩ D N
Remark 5.8 (Computational verification). Our implementation verified the existence of centered Goldbach partitions for all even integers 2 N 2N 2 N 8 ≤ 2 N ≤ 2 ⋅ 10 6 8 \le2N \le2 \cdot10^6 8 ≤ 2 N ≤ 2 ⋅ 1 0 6 Table 1 , Table 2 and Table 3 .
Conclusion
We have embedded the distinct-prime Goldbach condition C N ∩ D N ≠ ∅ C_N \cap D_N \ne\varnothing C N ∩ D N = ∅ N N N E N E_N E N N N N 2 N 2N 2 N C N ∩ D N ≠ ∅ C_N \cap D_N \ne\varnothing C N ∩ D N = ∅ M M M 2 N 2N 2 N
Summary of Main Results
Proposition 5.7 gives R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 Theorem 3.13 proves unconditionally that R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 X / 9600 X/9600 X /9600 N ≤ X N \le X N ≤ X X ≥ 10 6 X \ge10^6 X ≥ 1 0 6 Theorem 3.9 proves unconditionally that ∣ E N ∣ > N / 760 |E_N| > N/760 ∣ E N ∣ > N /760 N ≥ 3275 N \ge3275 N ≥ 3275 Proposition 3.11 , deduced from a theorem of Mikawa, gives ∣ A N ∖ E N ∣ ≪ D N / ( log N ) D |A_N \setminus E_N| \ll_D N/(\log N)^D ∣ A N ∖ E N ∣ ≪ D N / ( log N ) D N N N O D ( N / ( log N ) D ) O_D(N/(\log N)^D) O D ( N / ( log N ) D ) P < N P < N P < N 2 N − P 2N-P 2 N − P P ′ − P P' - P P ′ − P Theorem 3.1 establishes the geometric equivalence, and Proposition 3.2 gives the exact centered set-intersection criterion. Proposition 5.1 and Corollary 5.2 record what follows rigorously from Dusart’s short-interval theorem: the interval ( N , 2 N ) (N, 2N) ( N , 2 N ) E N E_N E N Lemma 2.6 , proved by differentiation, and Proposition 2.4 give Proposition 5.3 for every N ≥ 356 N \ge356 N ≥ 356 Lemma 3.3 shows C N ∩ D N ⊆ E N C_N \cap D_N \subseteq E_N C N ∩ D N ⊆ E N Lemmas 3.4 and 3.5 show that the prime gaps around N N N 2 N 2N 2 N E N E_N E N Theorem 5.4 shows that H ( N ) > 0 H(N) > 0 H ( N ) > 0 N N N ∣ A N ∖ E N ∣ |A_N \setminus E_N| ∣ A N ∖ E N ∣ Proposition 5.6 shows H R ( N ) < 0 H_R(N) < 0 H R ( N ) < 0 N ≥ 34 N \ge34 N ≥ 34 § 4 verifies R ( 2 N ) ≥ 1 R(2N) \ge1 R ( 2 N ) ≥ 1 H ( N ) > 0 H(N) > 0 H ( N ) > 0 4 ≤ N ≤ 10 6 4 \le N \le10^6 4 ≤ N ≤ 1 0 6 ∣ E N ∣ ≥ 0.9906 N |E_N| \ge0.9906N ∣ E N ∣ ≥ 0.9906 N 3275 ≤ N ≤ 10 6 3275 \le N \le10^6 3275 ≤ N ≤ 1 0 6 ( C N ∩ E N ) ∖ D N ≠ ∅ (C_N \cap E_N) \setminus D_N \ne\varnothing ( C N ∩ E N ) ∖ D N = ∅ N ≤ 2 14 N \le2^{14} N ≤ 2 14 N ∈ { 4 , 5 , 7 , 8 } N \in\{4, 5, 7, 8\} N ∈ { 4 , 5 , 7 , 8 }
Status of the Conjecture
The distinct-prime Goldbach conjecture is not proved in this paper. The paper proves that C N ∩ E N C_N \cap E_N C N ∩ E N C N C_N C N N N N N ≥ 356 N \ge356 N ≥ 356 H ( N ) > 0 H(N) > 0 H ( N ) > 0 Proposition 5.3 ). The step needed for a proof is the centering step: one must show that for some such M M M N + M N + M N + M
C N ∩ D N ≠ ∅ . C_N \cap D_N \ne\varnothing. C N ∩ D N = ∅ . Equivalently, by Proposition 5.7 , one must prove the positivity of R ( 2 N ) = ∣ C N ∩ D N ∣ R(2N) = |C_N \cap D_N| R ( 2 N ) = ∣ C N ∩ D N ∣ N > 2 ⋅ 10 18 N > 2 \cdot10^{18} N > 2 ⋅ 1 0 18
The Centered Gap
The proof gap is the centered correlation estimate R ( 2 N ) > 0 R(2N)>0 R ( 2 N ) > 0
H ( N ) = log 2 ( 2 N ) − ( ( N − 3 ) − ∣ E N ∣ ) H(N)=\log^{2}(2N)-((N-3)-|E_N|) H ( N ) = log 2 ( 2 N ) − (( N − 3 ) − ∣ E N ∣ ) measures half-difference coverage and is positive for every N ≤ 10 6 N\leq10^{6} N ≤ 1 0 6 H R ( N ) H_R(N) H R ( N ) N ≥ 34 N\geq34 N ≥ 34 Proposition 5.6 ). The prime-gap obstructions of Lemmas 3.4 and 3.5 force 26.98% of the missing values of E N E_N E N ( 2 19 , 2 20 ] (2^{19},2^{20}] ( 2 19 , 2 20 ] Table 4 ). A proof of the distinct-prime Goldbach conjecture along these lines requires ∣ C N ∩ D N ∣ ≥ 1 |C_N\cap D_N|\geq1 ∣ C N ∩ D N ∣ ≥ 1 N > 2 ⋅ 10 18 N>2\cdot10^{18} N > 2 ⋅ 1 0 18
Relation to the Classical Goldbach Conjecture
The framework addresses the variant requiring distinct primes, thus excluding 4 = 2 + 2 4=2+2 4 = 2 + 2 6 = 3 + 3 6=3+3 6 = 3 + 3 P = Q P=Q P = Q M = 0 M=0 M = 0
Open Questions
Several natural questions remain. By Proposition 3.11 , ∣ E N ∣ = N − o ( N ) |E_N|=N-o(N) ∣ E N ∣ = N − o ( N ) N N N c < 1 c<1 c < 1 N 0 N_0 N 0 ∣ E N ∣ ≥ c N |E_N|\geq cN ∣ E N ∣ ≥ c N N ≥ N 0 N\geq N_0 N ≥ N 0 Theorem 3.9 gives c = 1 / 760 c=1/760 c = 1/760 N 0 = 3275 N_0=3275 N 0 = 3275 ∣ E N ∣ ≥ 0.9906 N |E_N|\geq0.9906N ∣ E N ∣ ≥ 0.9906 N 3275 ≤ N ≤ 10 6 3275\leq N\leq10^{6} 3275 ≤ N ≤ 1 0 6 E N E_N E N Lemmas 3.4 and 3.5 do not force? Can one prove an explicit lower bound for R ( 2 N ) R(2N) R ( 2 N ) H ( N ) H(N) H ( N ) ∣ A N ∖ E N ∣ |A_N\setminus E_N| ∣ A N ∖ E N ∣ log 2 ( 2 N ) \log^{2}(2N) log 2 ( 2 N ) N N N Theorem 5.4 , the latter holds if p ′ − p ≥ 2 log 2 ( 2 p + 2 ) + 2 p'-p\geq2\log^{2}(2p+2)+2 p ′ − p ≥ 2 log 2 ( 2 p + 2 ) + 2 p < p ′ p<p' p < p ′ M = 0 M=0 M = 0
Statements and Declarations
Declaration of interests
The author declares no competing interests.
Funding sources
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Declaration of generative AI in scientific writing
The author used Claude Opus 5.5 to write the experiment scripts experiment.py and large_range.py, and to improve the readability of this manuscript; the author ran all the experiments. After using this tool, the author reviewed and edited the code and the content as needed and takes full responsibility for the content of the publication.
Data availability
Code and data: https://github.com/frankvegadelgado/goldbach ; the computations of Section 4 are reproduced by running the scripts experiment.py and large_range.py of the repository, which write their output to log files.
