On the Diophantine equation 36^x+p^y=z^2 where p is a prime number

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ศิริจันทร์ เวสารัชศาต
วนัสวี กุศล
ปวันรัตน์ ขาวดารา
พนธกร ภัคสุขพิมล

Abstract

This paper is to find all non-negative integer solutions gif.latex?(&space;x,&space;y,&space;z&space;,p) of the Diophantine equation  gif.latex?36^{x}&space;+&space;p^{y}=z^{2} where gif.latex?p เis a prime number. The result of this study found that, the Diophantine equation has non-negative integer solutions as follows gif.latex?(0,&space;3,&space;3,&space;2) , gif.latex?(1,&space;6,&space;10,&space;2) , gif.latex?(2,&space;6,&space;45,&space;3) and gif.latex?\left&space;(&space;k,&space;1,&space;6^{k}&space;+1,&space;2\cdot&space;6^{k}+1\right&space;) where gif.latex?k is a non-negative integer.

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How to Cite
1.
เวสารัชศาต ศ, กุศล ว, ขาวดารา ป, ภัคสุขพิมล พ. On the Diophantine equation 36^x+p^y=z^2 where p is a prime number. Prog Appl Sci Tech. [Internet]. 2019 Dec. 29 [cited 2024 Dec. 17];9(2):10-3. Available from: https://ph02.tci-thaijo.org/index.php/past/article/view/242922
Section
Mathematics and Applied Statistics

References

Alongkot Suvarnamani. Solutions of Diophantine equation . Int. J. of Mathematical Science and Applications. 2011. 1(3) : 1415-1419.

Somchit Chotchaisthit. On the Diophantine equation where is a prime number. American Journal of Mathematics and Sciences. 2012. 1(1) : 191-193.

Mihailescu, Primary cyclotomic units and a proof of Catalan’s conjecture. Journal of Reine Angew. Math. 2004. 572 : 167-195.