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Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a⁴ + b⁴ = c⁴. Part I

Advances in Pure Mathematics 2024年第4期14卷 303-319页

作者： Prosper Kouadio Kimou François Emmanuel Tanoé kouassi vincent kouakou Laboratoire d’ Informatique et de Math&#233matiques appliqu&#233es Institut Polytechnique F&#233lix Houphou&#235t BOIGNY Yamoussoukro Cote d&#8217Ivoire UFR Mathé matiques et Informatique Universit&#233 F&#233lix Houphouet BOIGNY Abidjan Cote d&#8217Ivoire UFR Sciences Fondamentales Appliqué es Universit&#233 NANGUI ABROGOUA Abidjan Cote d&#8217Ivoire

In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F0) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−10) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F0): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=

关键词： Factorisation in ℤ Greatest Common Divisor Pythagoras Equation Pythagorician Triplets Fermat's Equations Pythagorician Divisors Fermat's Divisors Diophantine Equations of Degree 2 4-Integral Closure of ℤ in ℚ

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New Asymptotic Results on Fermat-Wiles Theorem

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Advances in Pure Mathematics 2024年第6期14卷 421-441页

作者： Kimou Kouadio Prosper kouakou kouassi vincent Tanoé François UMRI MSN Felix Houphouet-Boigny National Polytechnic Institute Yamoussoukro Ivory Coast Nangui Abrogoua University Applied Fundamental Sciences Department Abidjan Ivory Coast UFRMI Flix Houphout-Boigny University Abidjan Ivory Coast

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

关键词： Fermat’s Last Theorem Fermat-Wiles Theorem Kimou’s Divisors Diophantine Quotient Diophantine Remainders Balzano Weierstrass Analysis Theorem

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Epidemiological, Diagnostic, Therapeutic and Progressive Aspects of Arterial Hypertension in Children at the University Hospital Center of Bouaké (Côte d’Ivoire)

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Open Journal of Pediatrics 2024年第2期14卷 379-390页

作者： Richard Azagoh-Kouadio Yao Kossonou Roland Yeboua kouassi Christian Yao John Patrick Yenan Iburaima Alamun Akanji Kouadio vincent Assé Pediatric Department of Angré University Hospital Abidjan Ivory Coast Pediatric Department of Bouaké University Hospital Bouaké Ivory Coast

Introduction: Arterial hypertension (AH) in children is under-diagnosed and often has a poor prognosis. The aim of this study was to describe the epidemiological, diagnostic, therapeutic and evolutionary aspects of hypertension in children at the University Hospital of Bouaké, with a view to improving the prognosis. Methods: This was a cross-sectional, analytical study carried out in the paediatrics department of Bouaké University Hospital. It concerned the medical records of children aged 4 to 15 hospitalised from 1 January 2017 to 31 December 2020 for hypertension. Diagnosis was based on the simplified blood pressure guidelines of the Expert Consensus of the French Society of Hypertension. The variables studied were epidemiological, diagnostic, therapeutic and evolutionary. Quantitative variables were compared at the significance level p ≤ 0.05. Results: The hospital incidence of hypertension was 0.32% (69/21,642). The sex ratio was 0.72. 97.1% of the children were over five years of age. Oedema (49.3%) and breathing difficulties (20.2%) were the main reasons for consultation. Hypertension was classified as threatening (56.5%), confirmed (31.9%) and borderline (11.6%). The cause was renal in 66.7%, dominated by impure nephrotic syndrome (24.6%). Treatment for hypertension consisted mainly of a diuretic (79.7%) and a calcium channel blocker (47.8%). Outcome was favourable in 50.7% of cases. Mortality was 20.3%. No factor was significantly associated with death. Conclusion: Hypertension in children at Bouaké University Hospital is serious. The aetiology is mainly renal. Early diagnosis and management are key to improving prognosis.

关键词： Hypertension Children Renal Pathologies Prognosis Côte d’Ivoire

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Assessment of the Vaccination Status of Children Aged 6 to 30 Months in a Tertiary Level Health Care Center in Côte d’Ivoire

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Open Journal of Pediatrics 2024年第3期14卷 536-546页

作者： Yenan John Patrick Kpangni Jean Bertrand Yao kouassi Christian Yeboua Yao Kossonou Roland Aka-Tanoh Koko Aude Hélène Avi-Siallou Christelle Honorine Asse Kouadio vincent Department of Pediatrics University Teaching Hospital of Bouake Bouake Ivory Coast Unit Pediatric Surgery University Teaching Hospital of Bouake Bouake Ivory Coast

Introduction: Vaccination coverage in Côte d’Ivoire over the period 2011 to 2015 was below the target of 95% for all antigens. The objective of this study was to analyze the vaccination status of children aged 6 to 30 months with a view to improving vaccination coverage. Patients and Methods: This was a descriptive cross-sectional study which took place from June to September 2018 in a tertiary health center, focusing on children aged 6 to 30 months with a correctly completed health record. The parameters studied were sex, age, educational level of mothers, dates of vaccine administration and reason for missed vaccination opportunities. Results: We retained 212 children. The sex ratio was 1.21 and 93% had received the BCG vaccine before the age of 1 month. The average ages of combined and co-administered vaccines for the 1st and 2nd doses were 7.66 ± 3.81 and 12.88 ± 3.95 weeks, respectively. The median was 16.57 weeks for the 3rd dose. The proportion of vaccinated subjects was greater than 90% for the BCG vaccine and the 3 doses of combined vaccines, and 77% for the yellow fever and measles vaccines. The reasons for non-vaccination were attributable to the children’s parents and health facilities. Conclusion: Improving vaccination coverage requires regular supply of vaccines to centers, and the involvement of all health professionals, community and religious leaders in the vaccination awareness process.

关键词： Children Vaccination Coverage Missed Opportunities

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Psychological Experience of Mothers of Children with Sickle Cell Disease Followed at the Pediatric Department of Bouaké University Teaching Hospital

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Open Journal of Pediatrics 2024年第1期14卷 149-163页

作者： Akanji Iburaima Alamun Aka-Tanoh Koko Aude Hélène Yao kouassi Christian Adou Leioh Romeo Sahi Gnantin Josette Landryse Amani Ehi Alexise Eleonore Boune Aboulaye Asse Kouadio vincent Paediatrics Department Bouaké University Teaching Hospital Bouaké C&#244te d’Ivoire

Introduction: Sickle cell disease has physical and emotional repercussions on the child and his family. The aim of this study was to describe the psychosocial experiences of mothers of children with sickle cell disease in order to improve the overall care of the child. Methods: This was a descriptive cross-sectional study carried out in the pediatrics department of Bouaké University Teaching Hospital from June to September 2023. It focused on mothers of major sickle-cell-affected children followed up in the pediatrics department of the Bouaké University Teaching Hospital. The variables studied were sociodemographic, psychological, social and economic. Results: Of the 40 mothers surveyed, 15% were not in school and 32.5% were unemployed. For them, sickle cell disease was of natural (genetic) origin in 90% and supernatural in 10%. They stated that the child had an average age of 36 months (extremes 7 and 108 months) when the disease was discovered. And 52% of them were satisfied with the way the disease was clearly and completely announced. Following the announcement, the questioned mothers said they had felt shock (35%), sadness (31.7%), guilt (23.3%) and discouragement (10%). Anxiety and depression were experienced by 77.5% and 22.5% respectively. In 60% of cases, they stated that the disease was incurable, and the outcome was fatal in 2.5% of cases. The child’s illness was a source of problems in the home in 25% of cases, represented by arguments in 92% and divorce in 8%. In 97.5% of cases, the mother told her family and friends about the child’s illness. In 90% of cases, the mother and child benefited from psychological support from family and friends. Conclusion: Sickle cell disease is a serious illness with a psychological and social impact on mothers. We recommend psychological support for mothers from the moment of diagnosis and throughout follow-up.

关键词： Psychosocial Experience Sickle Cell Disease Côte d’Ivoire

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Childhood Heart Disease in Côte d’Ivoire: Retrospective Multicentric Study about 228 Cases

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Open Journal of Pediatrics 2018年第4期8卷 375-388页

作者： Richard Azagoh-Kouadio kouakou Florent Diby Kouadio vincent Asse Loa Gnaba Koko Aude Aka-Tanoh Kossonou Roland Yeboua kouassi Christian Yao Christelle Avi Soumahoro Oulai Paediatric Ward of University Teaching Hospital of Treichville Abidjan Cote d’Ivoire Department of Cardiovascular and Thoracic Diseases CHU de Bouaké Bouaké Cote d’Ivoire Paediatric Ward of University Teaching Hospital of Bouake Bouaké Cote d’Ivoire

Introduction: In Cote d’Ivoire, the paediatrician has to face a diagnosis and management problem in front of pediatric heart disease, with consequent a high lethality. Objective: To describe the epidemiological, diagnostic, therapeutic and progressive aspects of childhood cardiopathy for the improvement of prognosis and professional practice. Materials and methods: This was a multicenter retrospective and descriptive study conducted from January 2011 to March 2016 in two main hospitals universities, one located in Abidjan and the other in Bouaké. It concerned children aged 0 to 15 admitted for cardiac disease diagnosed on clinical and/or echocardiographic arguments. The variables studied were the epidemiological, diagnostic, therapeutic and evolutionary aspects. Results: A total of 49.760 admissions including 228 infantile heart disease cases (congenital 113, acquired 51, indeterminate 64) are overall prevalence of 4.6‰. They involved 106 boys and 122 girls. In 74.6% of cases, the age was between 0 and 2 years. Respiratory distress 73.7% was the main reason for consultation. The main congenital heart diseases (CHD) are ventricular septal defect (VSD) (31%), atrial septal defect (ASD) (20.4%), atrioventricular canal (AVSD) (12.4%) and tetralogy of Fallot (TOF) (11.5%). As for acquired forms (AHD), rheumatic mitral insufficiency (41.2%) and tuberculous pericarditis (15.7%) were the two main causes. Treatment progress is marked by stabilization (71.1%) and death (14%). Death was significantly associated with low socioeconomic status (p = 0.01) and with complication (p 0.001). Conclusion: Infantile heart diseases are relatively less frequent and serious in Cote d’Ivoire. To improve the prognosis, close collaboration between the pediatrician and the cardiologist is required for early diagnosis and management.

关键词： Child Congenital Heart Disease Acquired Heart Disease Ivory Coast

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Type VI Aplasia Cutis Congenita: About a Case Report at University Teaching Hospital of Bouaké

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Open Journal of Pediatrics 2023年第1期13卷 146-152页

作者： Yao kouassi Christian Yeboua Yao Kossonou Roland Yenan John Patrick Akanji Iburaima Alamun Adou Leioh Romeo Sahi Gnantin Josette Landryse Amani Ehi Alexise Eleonore Avi-Siallou Christelle Honorine Aka-Tanoh Koko Aude Hélène Asse Kouadio vincent Department of Pediatrics Universiy Teaching Hospital of Bouaké Bouaké Ivory Coast

Introduction: Aplasia cutis congenita is a rare congenital dermatosis of which type VI represents the Bart’s syndrome. The aim of this case is to describe the epidemiological, clinical, therapeutic and prognostic characteristics of this condition in a country with limited resources, for the improvement of prognosis and professional practice. Observation: This is a eutrophic newborn, born at term by vaginal delivery, who presented at birth with a unilateral absence of skin on the anteromedial aspect of the right leg starting from the knee and extending to the medial aspect of the right foot, with a dystrophy of the nail of the right big toe without any other visible physical malformation. The evolution was marked at D3 of life by the appearance of bullae on the right hand and elbow as well as on the posterior aspect of the neck, making epidermolysis bullosa suspect. The mother was 38 years old, 8th gesture, 7th pare with history of consanguinity and collodion baby. The association of a localized congenital absence of skin on the lower limbs, epidermolysis bullosa and a nail anomaly led to the diagnosis of congenital cutaneous aplasia of type VI of Frieden’s classification or Bart’s syndrome. The evolution was satisfactory on the 7th day of life with the beginning of scarring. The management was medical. The outcome was unfavorable with the appearance of sepsis and hemorrhage leading to death. Conclusion: Although rare, the clinical diagnosis of Bart’s syndrome is simple. However, the management is complex and the prognosis is reserved. To improve this prognosis, the treatment must guarantee excellent control of the infectious and hemorrhagic risks, an adhesion and good therapeutic compliance by the parents and a rigorous monitoring.

关键词： Aplasia Cutis Congenita Bart’s Syndrome Newborn Prognosis Côte d’Ivoire

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Etiologies and Prognostic Factors of Dyspnea in Infants at the University Hospital Center (CHU) of Bouaké (Ivory Coast)

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Open Journal of Pediatrics 2023年第3期13卷 313-323页

作者： Yenan John Patrick Yeboua Yao Kossonou Roland Yao kouassi Christian Aka-Tanoh Koko Aude Hélène Akanji Iburaima Alamun Sahi Gnantin Josette Landryse Adou Leioh Romeo Amani Ehi Alexise Eleonore Avi-Siallou Christelle Honorine Asse Kouadio vincent Department of Pediatrics Universiy Teaching Hospital of Bouake Bouake Ivory Coast

Identify the epidemiological characteristics, etiologies and evolutionary aspects of dyspnea in infants. This was a retrospective study of infants hospitalized for dyspnea from January 1 to December 31, 2020. The parameters studied were sex, age, origin, vaccination status, existence of underlying pathology. Underlying, the diagnosis and the evolutionary modalities. Data analysis and processing were possible using Word, Excel and EPI info version 7 software. We retained 152 infants. The sex ratio was 1.34 and the median age was 4 months. Vaccines according to expanded immunization program (EPI) were up to date in 76.32%. The main antecedents with risk identified were malnutrition, hypotrophy at birth, interventricular communication. The pathologies observed were low acute respiratory diseases in 90.79%, ENT diseases in 04.60% and cardiac diseases in 03.95%. The median length of hospitalization was 4 days. Infants who died accounted for 15.13%. The median age of infants who died was 4 months. The median time to onset of death was 1.63 days. The risk factors for death were age < 6 months (p = 0.003;CI [1.27;9.33]), outdated vaccines (p = 0.012;CI [1.18;5.17]), history with risk (p = 0.031;CI [1.02;4.54]). Dyspnea in infants remains a concern in our service. Reducing mortality involves developing procedures for the management of lower respiratory ailments, continuous staff training and strengthening the technical platform.

关键词： Infants Dyspnea Prognostic Factors

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Medication Error (ME) in the Pediatrics Service of the University Teaching Hospital (CHU) of Bouaké(Côte d’Ivoire)

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Open Journal of Pediatrics 2018年第2期8卷 141-153页

作者： Kouadio vincent Asse Kossonou Roland Yeboua Richard Azagoh-Kouadio kouassi Christian Yao Koko Aude Aka-Tanoh Christelle Avi Mocket Adolphe Ehouman Kouie Jeannot Plo Paediatric Ward of University and Teaching Hospital of Bouake Bouaké Cote d’Ivoire Paediatric Ward of University and Teaching Hospital of Treichville Abidjan Cote d’Ivoire Olopam Pharma and Research & Development Abidjan Cote d’Ivoire

Introduction: Medication error (ME) is a common cause of childhood morbidity and mortality reported in developed countries. It has not yet been described in the pediatric services in Cote d’Ivoire. Objective: To evaluate ME in pediatrics for the improvement of professional practice. Materials and Methods: This was a cross-sectional, descriptive and analytical study conducted in the service of pediatrics at the University Teaching Hospital (CHU) of Bouaké from January 11 to February 25, 2016. It involved 204 children hospitalized and treated with drugs. The variables studied were the prescription, dispensing and administration of the drugs. Results: A total of 483 prescriptions notes were recorded for 733 therapeutic lines, i.e. 2.3 prescriptions notes and 3.6 lines per child. The overall prevalence of ME was 31% (prescription 83%, dispensing 0%, administration 11%). The prescription error concerned the illegible writing (20%), the omission of the stamp & signature (47%), the omission of the date (33%), weight (66%), age (25%) and name of the child (9%).The drug name and the dosage were incorrect in 99% and 1% respectively. Regarding the administration error, care was provided by a person without a nursing qualification in 65.5% of cases. On the patient’s care record, the care giver’s name was omitted in 100%, the child’s name in 57% and the date in 47%. The route of administration and the administered dose were each incorrect in 5%. The student nurses and assistant nurse caregivers committed more prescription errors than doctors and nurses and midwives (p= 0,000 X2 = 44.91). Administration error was not significantly related to the service, the socio-professional category, the gender and the age of the patient. Conclusion: EM is common in pediatrics at the University Teaching Hospital (CHU) of Bouaké. To prevent it, it is necessary to strengthen the staff’s capacity to prescribe and administer the drug in children.

关键词： Pediatrics Medication Error Hospitalization Ivory Coast

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Ascites of Great Abundance Revealing a Nephrotic Congenital Syndrome at the University Teaching Hospital of Bouaké: About a Case

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Open Journal of Pediatrics 2018年第1期8卷 19-24页

作者： Kouadio vincent Asse kouassi Christian Yao Richard Azagoh-Kouadio Serge Gohi Bi Irie Kossonou Roland Yeboua John Patrick Yenan Koko Aude Aka-Tanoh Christelle Avi Paediatric Ward of University and Teaching Hospital of Bouake Bouaké Cote d’Ivoire Paediatric Ward of University and Teaching Hospital of Treichville Abidjan Cote d’Ivoire Polyvalent Reanimation Ward of University and Teaching Hospital of Bouaké Bouaké Cote d’Ivoire

Congenital nephrotic syndrome (CNS) is defined as the presence of proteinuria > 50 mg/kg/24h associated with a protein concentration g/L or albuminemia 30 g/L in an infant less than 3 months old. The CNS is rare, of various clinical forms dominated by the Finnish type caused by a mutation of the NPHS1 gene located on chromosome 19. The edematous syndrome is the most common mode of discovery. We report a case discovered in an infant of 50 days admitted for ascites of great abundance. The aim of this study was to describe the main epidemiological, diagnostic, therapeutic and evolutionary aspects of this syndrome. Improving the prognosis of this condition requires advocacy with the political authorities of Cote d’Ivoire to provide Teaching Hospital for the resources needed to perform kidney transplantation.

关键词： Ascites Congenital Nephrotic Syndrome Finnish Type Renal Transplantation Cote d’Ivoire

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