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Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y

The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

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Publication Date
Sun Jun 30 2013
Journal Name
College Of Islamic Sciences
The Greatness of God in Creating the Hoopoe Bird - A Miracle Study - Explanatory

Praise be to God, Lord of the worlds, and prayers and peace be upon the best of the messengers, Muhammad, and all his family and companions. And after:
For when I had the pleasure of looking at the chapters of the Noble Qur’an, and its evident signs, my view fell on one of its brightest verses, and it is the Almighty saying: ﭽﯜ ﯝ ﯞ ﯟ ﯠ ﯡ ﯢ ﯣ ﯤ ﯥ ﯦ ﯧ ﯨﭼ (). The tag: (God’s Greatness in Creating a Hoopoe). The research methodology required that I divide it into four sections:
The first topic: an overview of birds.
The second topic: the characteristics and characteristics of hoopoe.
The third topic: the logic of the bird.
The fourth topic: Praise the bird.
This has relied in my research on

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Publication Date
Sun Jan 01 2017
Journal Name
International Journal Of Science And Nature
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Publication Date
Fri Mar 29 2024
Journal Name
مجلة العلوم التربوية والنفسية
Needs in the stories of the Brothers Grimm (a study in content analysis)

The current study aims to identify the needs in the stories of the Brothers Grimm. The research sample consisted of (3) stories, namely: 1- The story of the Thorn Rose (Sleeping Beauty) 2- The story of Snow White 3- The story of Little Red Riding Hood. The number of pages analyzed reached (15.5) pages, and to achieve the research objectives, Murray's classification of needs was adopted, which contains (36) basic needs that are further divided into (129) sub-needs. The idea was adopted as a unit of analysis and repetition as a unit of enumeration, Reliability was extracted in two ways: 1- Agreement between the researcher and himself over time, where the agreement coefficient reached 97%. The second was agreement between the researcher and tw

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Publication Date
Sat Dec 30 2023
Journal Name
Iraqi Journal Of Science
Influence of the Beam Size Radiation on the Depth Dose by Using 60Co

Radiotherapy is medical use of ionizing radiation, and commonly applied to the
cancerous tumor because of its ability to control cell growth.
The amount of radiation used in photon radiation therapy called dose (measured
in grey unit), which depend on the type and stage of cancer being treated.
In our work, we studied the dose distribution given to the tumor at different
depths (zero-20 cm) treated with different field size (4×4- 23×23 cm).
Results show that the deeper treated area has less dose rate at the same beam
quality and quantity. Also it has been noted increasing in the field increasing in the
depth dose at the same depth even if the radiation energy is constant. Increasing in
radiation dose attribut

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Publication Date
Thu Apr 01 2021
Journal Name
Iop Conference Series: Earth And Environmental Science
Defining the aspects of the local urban sustainability: Eco- cities as a model

The absence of ecological perception in the local urbanization resulted in the lack of a clear conception of achieving sustainability in its simplest form in the urban reality and in the city of Baghdad in particular. The research assumes the possibility of achieving urban sustainability in Iraqi cities by applying the cities for the most effective methods to implemented ecological solutions and introducing appropriate urban planning tools and improve the living environment. The research focuses on the ability to define some aspects to achieve a sustainable local urban identity from global experiences. This was performed by proposing a scheduled theoretical framework, through which the features of sustainability can be extrapolated from the

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Publication Date
Sat May 01 2021
Journal Name
Journal Of Physics: Conference Series
Anatomical histological structure of the cerebellum in the Iraqi frog Rana ridibunda ridibunda
Abstract<p>The current study was designed to investigate the histological structure of the cerebellum in the Iraqi frog <italic>Rana ridibunda ridibunda</italic>. The cerebellum was removed surgically from 20 Iraqi frogs, <italic>Rana ridibunda ridibunda</italic>, and preserved in fixative of 10% formalin solution for a period of 72 hr, then the histological sections were prepared by the method of paraffin. The results of the study showed that the cerebellum of the adult Iraqi frog is one of the rhombencephalon regions, and consists of three layers, molecular layer, purkinje cells layer, and granular layer. The present histological findings suggest that the cerebellum of the </p> ... Show More
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Publication Date
Mon Oct 31 2022
Journal Name
Periodicals Of Engineering And Natural Sciences (pen)
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Publication Date
Thu Apr 01 2021
Journal Name
Iop Conference Series: Earth And Environmental Science
Defining the aspects of the local urban sustainability: Eco-cities as a model
Abstract<p>The absence of ecological perception in the local urbanization resulted in the lack of a clear conception of achieving sustainability in its simplest form in the urban reality and in the city of Baghdad in particular. The research assumes the possibility of achieving urban sustainability in Iraqi cities by applying the cities for the most effective methods to implemented ecological solutions and introducing appropriate urban planning tools and improve the living environment. The research focuses on the ability to define some aspects to achieve a sustainable local urban identity from global experiences. This was performed by proposing a scheduled theoretical framework, through which th</p> ... Show More
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Publication Date
Mon Mar 01 2021
Journal Name
Journal Of Physics: Conference Series
The Bifurcation Analysis and Persistence of the Food Chain Ecological Model with Toxicant
Abstract<p>In this work, the occurrence conditions of both local Bifurcation and persistence were studied, Saddle-node bifurcation appears near fourth point, near the first point, the second point and the third point a transcritical bifurcation occurred but no pitchfork bifurcation happened near any of the four equilibrium points. In addition to study conditions for Hopf-bifurcation near positive stable point that is the fourth point. Besides discuss persistence occurrence as globally property of the food chain of three species include prey, first predator and top predator with impact of toxin in all species and harvesting effect on the predator’s only. Numerical results for the set of hypothe</p> ... Show More
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Publication Date
Sun Sep 24 2023
Journal Name
Arab World English Journal
The Correlation between EFL Learners’ Academic Intelligence and the Level of Productive Skills

Communicating effectively by gaining productive skills in a classroom setting is one of the critical goals of learning the English language. The current study was conducted to explore the correlation of EFL learners’ level of academic intelligence with their productive skills. The study tries to find an answer to what is the correlation between EFL learners’ academic intelligence and level of production skills. The study population represents EFL students at the departments of English language of the Iraqi Colleges of Education for the academic year (2022-2023). The sample includes 310 EFL students selected from the 3rd year of the Department of English of the College of Education, Ibn-Rushd for Human Sciences/University of Bagh

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