Finding a perfect triangle was stated as an open problem by Guy in 6. Numerous researches have been done in the past to find such a triangle, unfortunately, to date, no one has ever found one, nor ...has proved its non-existence. However, on the bright side, there are partial results which show that there exist triangles that satisfy five or even six of the seven parameters to be rational. In this paper, we perform an extensive search to investigate if we can extract any perfect triangles from the curve C4 based on the final unsolved case in 9, which will then complete the proof of existence or non-existence of perfect triangle on the curve. Multiple conjectures were tested to eliminate the possibilities of finding a perfect triangle from the last unsolved case of n ≡ 3024 (mod 6052) in 9. Finally, a theorem was proved, which was subtle enough to eliminate this case, proving that there does not exist any perfect triangle arising from the curve C4.
Abstract
Background and Objectives
Grandparents are key resources in grandchildren care globally. However, mixed findings indicated that multiple role engagement may enhance well-being and bring ...demands on grandparent caregivers in different contexts. This systematic review examines the association between the intensity of grandparent caregiving and their health and well-being (i.e., physical, mental, cognitive, and life satisfaction) by continent and country/region.
Research Design and Methods
Systematic searches were conducted in 4 databases. Peer-reviewed articles with quantitative designs published between 1990 and November 2021 were identified. A rigorous selection process was followed using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines. The studies were critically appraised, and their results were narratively synthesized.
Results
Sixty-five articles from 29 countries/regions were included. Findings suggested a concave curvilinear relationship between the intensity of grandparent caregiving and their health and well-being, with the optimal caregiving intensity varying across sociocultural contexts. In Europe, Oceania, the Middle East, and South America, providing supplementary or occasional care seems beneficial for grandparents’ health and well-being, especially supporting dual-earner families. In East Asia, economic resources appear to buffer the adverse effect of primary care on grandparents’ well-being. In the United States, findings vary across ethnicity/race.
Discussion and Implications
Collectively, the intensity of grandparent caregiving, health, and well-being is complicated by grandparents’ roles in the family and cultural differences. Acknowledging the bidirectional relationship between well-being and grandparents’ capacity for providing care, the well-being as outcome is a limitation. Despite so, this systematic review calls for culturally-tailored family programs to support grandparent caregiving.
The present study aims to identify, whether and how supplementary grandchild care is causally related to grandparents' self-rated health (SRH). Based on longitudinal data drawn from the German Aging ...Survey (DEAS; 2008–2014), I compare the results of pooled OLS, pooled OLS with lagged dependant variables (POLS-LD), random and fixed effects (RE, FE) panel regression. The results show that there is a positive but small association between supplementary grandchild care and SRH in POLS, POLS-LD, and RE models. However, the fixed effects model shows that the intrapersonal change in grandchild care does not cause a change in grandparents' SRH. The FE findings indicate that supplementary grandchild care in Germany does not have a causal impact on grandparents' SRH, suggesting that models with between-variation components overestimate the influence of grandchild care on grandparents’ health because they do not control for unobserved (time-constant) heterogeneity.
•The article examines health consequences for grandparents providing grandchild care.•The article focuses on supplementary grandchild care and Self-Rated Health.•The article compares different panel model approaches with each other.•Controlling for unobserved (time-constant) heterogeneity leads to null-effect results.
This study examined the cross-sectional associations between intergenerational caregiving and health risks among sandwiched Chinese grandparents who provide care to grandchildren, great-grandparents, ...or both. Drawing on biomarker data from the 2011 wave of the China Health and Retirement Longitudinal Study (N range = 2,189–3,035), we measured age-related biological health risks of hypertension, diabetes, inflammation, and allostatic load. We found that health risks did not necessarily increase with the intensity of intergenerational caregiving. Providing care to grandchildren and great-grandparents simultaneously was not as detrimental to health as reported in earlier studies from the United States. Sandwiched grandparents could benefit from providing care to grandchildren or great-grandparents only. These unexpected findings might be related to the cultural mandates of filial piety and family solidarity in China. Grandfathers and grandmothers experienced different associations between varying types of intergenerational caregiving and health risks.
We know little about custodial grandparents of children with autism spectrum disorder (ASD) who offer a vital social safety net. 117 custodial grandparents of children with ASD from 37 states ...completed an online survey with open-ended questions about their “greatest challenges and joys” as grandparent. Grounded theory analysis revealed four categories of experience (Issues with Adult Children, Caregiving Burden, Coping, & Wisdom) explained by 15 themes. Grandparents’ stressors encompassed custody issues, ASD problem behaviors like tantrums and eloping, insufficient ASD services, financial burden, 24/7 caregiving demands, social isolation, and fears for the future. Grandparents’ coping included celebrations of progress, unconditional love, faith, and a positive focus. Grandparents’ wisdom included patience and insight. Recommendations to support these caregivers are provided.
Background. Latino grandparents are often involved in rearing of grandchildren. However, their roles in promoting Latino children’s leisure time physical activity (LTPA) have been largely overlooked ...by researchers. Aim. The purpose of this study is to examine Latino grandparents’ involvement in and support for grandchildren’s (age 2-12) LTPA as well as issues facing Latino grandparents. Method. Participants included 53 Latino grandparents who lived with/cared for or used to live with/care for grandchildren of 2 to 12 years old in three low-income areas of Los Angeles County. Data were collected using 4 semistructured focus groups and 26 semistructured interviews. Transcribed data were systematically and iteratively coded and analyzed. Results. Latino grandparents perceived various benefits of physical activity to their grandchildren’s physical and mental health, and supported grandchildren’s LTPA directly and indirectly. The direct support included doing physical activity with grandchildren, taking grandchildren to places for physical activities, verbally asking grandchildren to do physical activities, and rewarding grandchildren for doing physical activities. The indirect support was made through providing suggestions to parents. Mothers usually regulated Latino grandparents’ involvement in or support for grandchildren’s LTPA and tended to promote a maternal lineage. Some Latino grandparents reported barriers to supporting grandchildren’s LTPA, including lack of information, access to affordable and safe LTPA facilities/programs, transportation, money, and time. Conclusions. Latino grandparents are advocates for grandchildren’s LTPA and support grandchildren’s LTPA directly and indirectly. It may be promising to develop interventions to increase Latino grandparents’ competency in supporting grandchildren’s LTPA.
In this manuscript, we update the literature over the last decade in addressing several new content areas that have emerged in the grandfamilies literature, along with issues that are still important ...to understanding grandparents raising their grandchildren today.
The social science and gerontological literature since 2004 was accessed, reviewed, organized topically, and integrated, based upon an exhaustive PsychINFO literature search.
Our review indicates an ongoing and/or growing emphasis on (a) the strengths of grandparent raising grandchildren, (b) diversity among grandfamilies along a number of parameters, (c) the social-interpersonal, cultural, and policy-related contexts of grandfamilies, (d) process-focused research, (e) parenting, parenting skills, and family relationships, (f) grandparent psychological distress, (g) targets for and the efficacy of interventions with grandfamilies, and (h) methodological issues relevant to the study of grandfamilies.
We discuss the implications of our findings in terms of more completely understanding grandfamilies along a number of parameters, as well as presenting specific recommendations for future research and practice.
The purpose of this systematic review and meta-analysis was to determine if raising grandchildren is related to diminished mental health well-being in custodial grandparents compared to ...contemporaries who are not raising grandchildren. Relevant studies were identified via comprehensive literature searches of electronic databases. We included six studies in the meta-analysis. A random effects model was used to calculate effect sizes. The results of the meta-analysis yielded a statistically significant, small-to-moderate summary effect size (ES) indicating caregiving grandparents generally experience a larger degree of adverse mental health outcomes than their non-parenting counterparts. While previous studies have produced conflicting results regarding the mental health outcomes experienced by custodial grandparents, the present meta-analysis findings provide another level of evidence that substantiates their emotional vulnerability. Recommendations for subsequent research are discussed.
The COVID-19 pandemic has exposed the vulnerability of many families, including grandparent kinship families, to deal with a health/economic crisis. The fear of COVID-19 plus stay-at-home orders have ...increased individuals’ psychological distress. Moreover, school closures and homeschooling further increased parenting stress among caregivers.
This study examined the relationship between material hardship and parenting stress among grandparent kinship providers, and assessed grandparents’ mental health as a potential mediator to this relationship during the COVID-19 pandemic in the United States.
Grandparent kinship providers (N = 362) that took primary care of their grandchildren participated in a cross-sectional survey via Qualtrics Panels in June 2020 in the United States.
Descriptive and bivariate analyses, binary logistic regression, and mediation analyses were conducted using STATA 15.0.
Suffering material hardship was significantly associated with higher odds of experiencing parenting stress among grandparent kinship providers, and grandparents’ mental health partially mediated this association.
Addressing material and mental health needs among grandparent kinship providers is critical to decreasing their parenting stress.
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