Wipawee Tangjai, Chusak Chubthaisong, "On the Diophantine equation 3x+py=z2 where p ≡ 2 (mod 3)
," WSEAS Transactions on Mathematics, vol. 20, pp. 283-287, 2021, DOI:10.37394/23206.2021.20.29
Wipawee Tangjai, Chusak Chubthaisong. On the Diophantine equation 3x+py=z2 where p ≡ 2 (mod 3)
.
WSEAS Transactions on Mathematics. 2021;20:283-287. 10.37394/23206.2021.20.29
WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 20, 2021
On the Diophantine equation 3x+py=z2 where p ≡ 2 (mod 3)
Authors: ,
Abstract: Let p be a prime number where p ≡ 2 (mod 3). In this work, we give a nonnegative integer solution for
the Diophantine equation 3<sup>x</sup>+p<sup>y</sup>=z<sup>2</sup>
. If y = 0, then (p, x, y, z) = (p, 1, 0, 2) is the only solution of the equation
for each prime number p. If y is not divisible by 4, then the equation has a unique solution (p, x, y, z) = (2, 0, 3, 3).
In case that y is a positive integer that is not divisible by 4, we give a necessary condition for an existence of a
solution and give a computational result for p < 10<sup>17</sup>. We also give a necessary condition for an existence of a solution for q<sup>x</sup> + p<sup>y</sup>=z<sup>2</sup> when p and q are distinct prime numbers.
Pages: 283-287 WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 20, 2021, Art. #29 DOI: 10.37394/23206.2021.20.29
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