From 0b994218484896e866ad3e8cbe132375388ea511 Mon Sep 17 00:00:00 2001 From: Andrew Yoo <199436365+andrew-yoo@users.noreply.github.com> Date: Wed, 17 Jun 2026 01:29:47 -0400 Subject: [PATCH 1/5] Update 45a.md --- constants/45a.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/constants/45a.md b/constants/45a.md index d8312e7..99d01dc 100644 --- a/constants/45a.md +++ b/constants/45a.md @@ -19,13 +19,14 @@ $$ | $0.490941$ | [HR2006] | | | $0.490341088858244$ | [CDL2024] | | | $<0.490249407811155$ | [G2026] | 36-prime finite obstruction certificate with exact cluster-update verification. | +| $<0.49024898035099$ | [Y2026] | | ## Known lower bounds | Bound | Reference | Comments | | ----- | --------- | -------- | -| 0 | Trivial | | -| >0 | [R1934] | | +| $0$ | Trivial | | +| $>0$ | [R1934] | | | $0.0868$ | [CS2004] | | | $0.0936$ | [P2006] | | | $0.093627$ | [HS2010] | | @@ -88,6 +89,7 @@ The external verifier independently recomputes the 13-prime seed histogram from - [G2026] Griego, Sebastian. 36-prime obstruction certificate for Romanoff's constant upper-density bound, 2026. [Code and verification](https://github.com/sebastian-griego/c45-romanoff-certificate/tree/v1-c45-certificate). - [G2026-ub-0-490249407811155] **loc:** external verifier repository, `verify_all.sh`. **quote:** "The exact rational verifier proves the upper-density bound \(\overline d(A)<0.490249407811155\)." +- [Y2026] Yoo, Andrew, "An improved upper bound on Romanoff's constant," 2026. https://github.com/andrew-yoo/romanoff. ## Contribution notes From 833d93d562992f3db711354bc23fcc44d9b356f5 Mon Sep 17 00:00:00 2001 From: Andrew Yoo <199436365+andrew-yoo@users.noreply.github.com> Date: Wed, 17 Jun 2026 01:31:20 -0400 Subject: [PATCH 2/5] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 074879d..6dbf220 100644 --- a/README.md +++ b/README.md @@ -77,7 +77,7 @@ Bounds for which the level of available verification is currently at minimal lev | [42](https://teorth.github.io/optimizationproblems/constants/42a.html) | Turan's pure power sum constant | 0.5 | 0.69368 | | [43](https://teorth.github.io/optimizationproblems/constants/43a.html) | Gilbert-Pollak conjecture (Steiner ratio) | 0.8559 | 0.86602540378 | | [44](https://teorth.github.io/optimizationproblems/constants/44a.html) | Maximal number of relevant variables in degree-$d$ Boolean functions | 1.5 | 4.394 | -| [45](https://teorth.github.io/optimizationproblems/constants/45a.html) | Density of odd integers that are the sum of a prime and a power of two | 0.107648 | 0.490341088858244 | +| [45](https://teorth.github.io/optimizationproblems/constants/45a.html) | Density of odd integers that are the sum of a prime and a power of two | 0.107648 | <0.49024898035099 | | [46](https://teorth.github.io/optimizationproblems/constants/46a.html) | Fourier restriction constant for the 2-sphere | 3 | $\frac{22}{7}\approx 3.142857$ | | [47](https://teorth.github.io/optimizationproblems/constants/47a.html) | Centered Hardy-Littlewood maximal constant in dimension $2$ | $\frac{11+\sqrt{61}}{12}\approx 1.5675208$ | 4 | | [48](https://teorth.github.io/optimizationproblems/constants/48a.html) | One-dimensional convex sub-Gaussian comparison constant | $\approx 5.33386$ | $\approx 5.33386$ | From 95ebe730b473c688fc7e4e1e1464d02fc86e5a0f Mon Sep 17 00:00:00 2001 From: Andrew Yoo <199436365+andrew-yoo@users.noreply.github.com> Date: Fri, 19 Jun 2026 18:10:04 -0400 Subject: [PATCH 3/5] Update 45a.md --- constants/45a.md | 8 +------- 1 file changed, 1 insertion(+), 7 deletions(-) diff --git a/constants/45a.md b/constants/45a.md index 99d01dc..3e507f4 100644 --- a/constants/45a.md +++ b/constants/45a.md @@ -4,12 +4,6 @@ $C_{45}$ is the asymptotic density (if it exists) of the set of odd integers that can be expressed as the sum of a prime number and a power of two. -Writing $A$ for this set, an explicit finite obstruction computation gives the upper-density bound -$$ -\overline d(A)<0.490249407811155. -$$ -[G2026-ub-0-490249407811155] If the density exists, this gives the corresponding upper bound for $C_{45}$. - ## Known upper bounds | Bound | Reference | Comments | @@ -89,7 +83,7 @@ The external verifier independently recomputes the 13-prime seed histogram from - [G2026] Griego, Sebastian. 36-prime obstruction certificate for Romanoff's constant upper-density bound, 2026. [Code and verification](https://github.com/sebastian-griego/c45-romanoff-certificate/tree/v1-c45-certificate). - [G2026-ub-0-490249407811155] **loc:** external verifier repository, `verify_all.sh`. **quote:** "The exact rational verifier proves the upper-density bound \(\overline d(A)<0.490249407811155\)." -- [Y2026] Yoo, Andrew, "An improved upper bound on Romanoff's constant," 2026. https://github.com/andrew-yoo/romanoff. +- [Y2026] Yoo, Andrew, "An improved upper bound on Romanoff's constant," 2026. https://github.com/andrew-yoo/romanoff/tree/v1. ## Contribution notes From cc89e5c597a19f453d0d2ad647e3f591075335db Mon Sep 17 00:00:00 2001 From: Andrew Yoo <199436365+andrew-yoo@users.noreply.github.com> Date: Sat, 20 Jun 2026 01:40:07 -0400 Subject: [PATCH 4/5] Update 45a.md --- constants/45a.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/constants/45a.md b/constants/45a.md index 3e507f4..57b0d03 100644 --- a/constants/45a.md +++ b/constants/45a.md @@ -13,7 +13,7 @@ $C_{45}$ is the asymptotic density (if it exists) of the set of odd integers tha | $0.490941$ | [HR2006] | | | $0.490341088858244$ | [CDL2024] | | | $<0.490249407811155$ | [G2026] | 36-prime finite obstruction certificate with exact cluster-update verification. | -| $<0.49024898035099$ | [Y2026] | | +| $<0.490248930138966$ | [Y2026] | | ## Known lower bounds @@ -83,7 +83,7 @@ The external verifier independently recomputes the 13-prime seed histogram from - [G2026] Griego, Sebastian. 36-prime obstruction certificate for Romanoff's constant upper-density bound, 2026. [Code and verification](https://github.com/sebastian-griego/c45-romanoff-certificate/tree/v1-c45-certificate). - [G2026-ub-0-490249407811155] **loc:** external verifier repository, `verify_all.sh`. **quote:** "The exact rational verifier proves the upper-density bound \(\overline d(A)<0.490249407811155\)." -- [Y2026] Yoo, Andrew, "An improved upper bound on Romanoff's constant," 2026. https://github.com/andrew-yoo/romanoff/tree/v1. +- [Y2026] Yoo, Andrew, "An improved upper bound on Romanoff's constant," 2026. https://github.com/andrew-yoo/romanoff/tree/v2. ## Contribution notes From d0b9e5c3f72ad1c5d589a493c3f15c7e7a7f3fea Mon Sep 17 00:00:00 2001 From: Andrew Yoo <199436365+andrew-yoo@users.noreply.github.com> Date: Sat, 20 Jun 2026 01:40:31 -0400 Subject: [PATCH 5/5] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 10aca14..cfeef46 100644 --- a/README.md +++ b/README.md @@ -77,7 +77,7 @@ Bounds for which the level of available verification is currently at minimal lev | [42](https://teorth.github.io/optimizationproblems/constants/42a.html) | Turan's pure power sum constant | 0.5 | 0.69368 | | [43](https://teorth.github.io/optimizationproblems/constants/43a.html) | Gilbert-Pollak conjecture (Steiner ratio) | 0.8559 | 0.86602540378 | | [44](https://teorth.github.io/optimizationproblems/constants/44a.html) | Maximal number of relevant variables in degree-$d$ Boolean functions | 1.5 | 4.394 | -| [45](https://teorth.github.io/optimizationproblems/constants/45a.html) | Density of odd integers that are the sum of a prime and a power of two | 0.107648 | <0.49024898035099 | +| [45](https://teorth.github.io/optimizationproblems/constants/45a.html) | Density of odd integers that are the sum of a prime and a power of two | 0.107648 | <0.490248930138966 | | [46](https://teorth.github.io/optimizationproblems/constants/46a.html) | Fourier restriction constant for the 2-sphere | 3 | $\frac{22}{7}\approx 3.142857$ | | [47](https://teorth.github.io/optimizationproblems/constants/47a.html) | Centered Hardy-Littlewood maximal constant in dimension $2$ | $\frac{11+\sqrt{61}}{12}\approx 1.5675208$ | 4 | | [48](https://teorth.github.io/optimizationproblems/constants/48a.html) | One-dimensional convex sub-Gaussian comparison constant | $\approx 5.33386$ | $\approx 5.33386$ |