A man in his 60s with multiple vascular comorbidities presented with sudden, painless vision loss in one eye. Although he had a high risk for atherosclerotic events, initial evaluation for stroke was negative for acute ischemia, but found to have markedly elevated inflammatory markers. Accordingly, giant cell arteritis was investigated and Ophthalmologic findings and fulfillment of the 2022 American College of Rheumatology/European Alliance of Associations for Rheumatology classification criteria supported the diagnosis of giant cell arteritis, despite a negative temporal artery biopsy. Management included high-dose glucocorticoids and delayed tocilizumab initiation due to the need for multiple vascular surgeries. Vision loss was irreversible, but systemic symptoms resolved and vascular interventions were successful. This case highlights the diagnostic and management complexities of biopsy-negative giant cell arteritis in patients with severe atherosclerotic vascular disease, emphasizing the importance of clinical judgment and established classification criteria when imaging and biopsy results are inconclusive.

Background: Dysphagia is a common functional impairment in elderly populations, often leading to severe complications such as malnutrition and aspiration pneumonia, significantly increasing healthcare burdens. Currently, effective prognostic assessment tools are lacking. The absolute lymphocyte count (ALC), a biomarker reflecting immune-nutritional status, has potential predictive value in this context, though its role in dysphagia prognosis remains unclear. Methods: This retrospective cohort study included 253 dysphagic patients who received percutaneous endoscopic gastrostomy (PEG) or total parenteral nutrition (TPN) between 2014 and 2017. Five patients with missing ALC were excluded. Cox regression models assessed the association between ALC and mortality. ALC was analyzed as both continuous variable (using restriocted cubic splines) and categorical tertiles, with additional threshold analyses to assess non-linearity. Kaplan–Meier survival curves and subgroup analyses were also performed. Results: Lower ALC was associated with poorer nutritional status, higher inflammatory markers, and greater comorbidity burden. Higher ALC was independently associated with reduced mortality (adjusted HR: 0.60; 95% CI: 0.44–0.83; p = 0.002). Patients in the highest tertile had significantly better survival than those in the lowest (HR: 0.37; 95% CI: 0.23–0.59; P < 0.001). A non-linear threshold effect was identified at ALC = 1.899×10⁹/L (p for non-linearity = 0.009). Kaplan–Meier analysis confirmed improved survival with higher ALC (p < 0.0001). Subgroup analyses showed the protective effect of higher ALC was consistent across age, sex, BMI, PEG use, and comorbidity strata, with no significant interactions. Conclusions: ALC is an independent, non-linear predictor of mortality in older dysphagic patients and may aid clinical risk stratification across diverse patient subgroups.

Legionnaires' disease became widely known following an outbreak of pneumonia in the United States in 1976. It is often caused by infection from artificial water sources such as cooling towers, water supply and heating systems, and recirculating hot tubs. To effectively implement infection prevention measures for Legionnaires' disease, collaboration among healthcare workers, water supply and heating system managers, building hygiene personnel, and other relevant parties is essential. It is important to note that outbreaks of Legionnaires' disease continue to occur frequently both domestically and internationally. While the number of reported cases of Legionnaires' disease in Japan has increased, the mortality rate has decreased but has stabilized at a lower level. Caution is also required as reports have been made in association with disasters and travel, in addition to artificial environmental water.

A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer $E_n\ge 4$, the residual set $ℛ\left(E_n\right)=\{E_n-p\mid$ $p<E_n,p\in \mathbb{P}\}$ is defined, and the universal residual set ${ℛ}_{\mathbb{E}}={{{\displaystyle \cup }}^{\text{}}}_{E_n}ℛ\left(E_n\right)$ is constructed. It is shown that ${ℛ}_{\mathbb{E}}$ contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies $G\left(E\right)\ge 2$ for all $E\ge 8$, and that the cumulative partition count satisfies ${\displaystyle \sum }_{E\le N}G\left(E\right)\gg \frac{N^2}{{\text{log}}^{4}N}$. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition $E_n=p+q$ is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer $n\ge 7$ can be expressed as $n=p+2q$, where $p,q\in \mathbb{P}$. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all $n\ge 9$. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence.