On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2
The Diophantine equation ax+py = z2 where p is prime is widely studied by many mathematicians. Solving equations of this type often include Catalan’s conjecture in the process of proving these equations. Here, we study the non-negative integer solutions for some Diophantine equations of such fami...
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| Formato: | Tese |
| Idioma: | English |
| Publicado em: |
2022
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| Acesso em linha: | http://eprints.usm.my/59441/1/AMR%20MOUSTAFA%20MOHAMED%20ALY%20ELSAYED%20ELSHAHED%20-%20TESIS%20cut.pdf |
| _version_ | 1825908087923408896 |
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| author | Elshahed, Amr Moustafa Mohamed Aly Elsayed |
| author_facet | Elshahed, Amr Moustafa Mohamed Aly Elsayed |
| author_sort | Elshahed, Amr Moustafa Mohamed Aly Elsayed |
| collection | USM |
| description | The Diophantine equation ax+py = z2 where p is prime is widely studied by many
mathematicians. Solving equations of this type often include Catalan’s conjecture in
the process of proving these equations. Here, we study the non-negative integer solutions
for some Diophantine equations of such family. We will use Mihailescu’s theorem
(which is the proof of Catalan’s conjecture) and elementary methods to solve the
Diophantine equations 16x −7y = z2, 16x − py = z2 and 64x − py = z2, then we will
study a generalization where (4n)x − py = z2 and x, y, z,n are non-negative integers.
By using Mihailescu’s theorem and a fundamental approach in the theory of numbers,
namely the theory of congruence, we will determine the solution of the Diophantine
equations 7x+11y = z2, 13x+17y = z2, 15x+17y = z2 and 2x+257y = z2 where x, y
and z are non-negative integers. Also, we will prove that for any non-negative integer
n, all non-negative integer solutions of the Diophantine equation 11n8x+11y = z2 are
of the form (x, y, z) = (1,n,3(11)
n2
) where n is even, and has no solution when n is
odd. Finally, we will concentrate on finding the solutions of the Diophantine equation
3x+ pmny = z2 where y = 1,2 and p > 3 a prime number. |
| first_indexed | 2024-03-06T16:13:46Z |
| format | Thesis |
| id | usm.eprints-59441 |
| institution | Universiti Sains Malaysia |
| language | English |
| last_indexed | 2024-03-06T16:13:46Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | usm.eprints-594412023-09-27T01:46:41Z http://eprints.usm.my/59441/ On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2 Elshahed, Amr Moustafa Mohamed Aly Elsayed QA1-939 Mathematics The Diophantine equation ax+py = z2 where p is prime is widely studied by many mathematicians. Solving equations of this type often include Catalan’s conjecture in the process of proving these equations. Here, we study the non-negative integer solutions for some Diophantine equations of such family. We will use Mihailescu’s theorem (which is the proof of Catalan’s conjecture) and elementary methods to solve the Diophantine equations 16x −7y = z2, 16x − py = z2 and 64x − py = z2, then we will study a generalization where (4n)x − py = z2 and x, y, z,n are non-negative integers. By using Mihailescu’s theorem and a fundamental approach in the theory of numbers, namely the theory of congruence, we will determine the solution of the Diophantine equations 7x+11y = z2, 13x+17y = z2, 15x+17y = z2 and 2x+257y = z2 where x, y and z are non-negative integers. Also, we will prove that for any non-negative integer n, all non-negative integer solutions of the Diophantine equation 11n8x+11y = z2 are of the form (x, y, z) = (1,n,3(11) n2 ) where n is even, and has no solution when n is odd. Finally, we will concentrate on finding the solutions of the Diophantine equation 3x+ pmny = z2 where y = 1,2 and p > 3 a prime number. 2022-07 Thesis NonPeerReviewed application/pdf en http://eprints.usm.my/59441/1/AMR%20MOUSTAFA%20MOHAMED%20ALY%20ELSAYED%20ELSHAHED%20-%20TESIS%20cut.pdf Elshahed, Amr Moustafa Mohamed Aly Elsayed (2022) On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2. Masters thesis, Universiti Sains Malaysia. |
| spellingShingle | QA1-939 Mathematics Elshahed, Amr Moustafa Mohamed Aly Elsayed On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2 |
| title | On The Solvability Of Some
Diophantine Equations Of The Form
ax+by = z2 |
| title_full | On The Solvability Of Some
Diophantine Equations Of The Form
ax+by = z2 |
| title_fullStr | On The Solvability Of Some
Diophantine Equations Of The Form
ax+by = z2 |
| title_full_unstemmed | On The Solvability Of Some
Diophantine Equations Of The Form
ax+by = z2 |
| title_short | On The Solvability Of Some
Diophantine Equations Of The Form
ax+by = z2 |
| title_sort | on the solvability of some diophantine equations of the form ax by z2 |
| topic | QA1-939 Mathematics |
| url | http://eprints.usm.my/59441/1/AMR%20MOUSTAFA%20MOHAMED%20ALY%20ELSAYED%20ELSHAHED%20-%20TESIS%20cut.pdf |
| work_keys_str_mv | AT elshahedamrmoustafamohamedalyelsayed onthesolvabilityofsomediophantineequationsoftheformaxbyz2 |