TY - JOUR
T1 - A new conjecture concerning the Diophantine equation x2 + by = cz
AU - Cao, Zhenfu
AU - Dong, Xiaolei
AU - Li, Zhong
PY - 2002/12
Y1 - 2002/12
N2 - In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x2 + by = cz has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1.
AB - In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if a = |Vr|, b = |Ur|, c = m2 + 1, and b ≡ 3 (mod 4) is a prime power, then the Diophantine equation x2 + by = cz has only the positive integer solution (x,y, z) = (a, 2, r), where r > 1 is an odd integer, m ∈ N with 2 | m and the integers Ur, Vr satisfy (m + √-1)r = Vr + Ur√-1.
KW - Exponential Diophantine equation
KW - Gauss integer
KW - Lucas sequence
KW - Primitive divisor
UR - https://www.scopus.com/pages/publications/0038326950
U2 - 10.3792/pjaa.78.199
DO - 10.3792/pjaa.78.199
M3 - 文章
AN - SCOPUS:0038326950
SN - 0386-2194
VL - 78
SP - 199
EP - 202
JO - Proceedings of the Japan Academy Series A: Mathematical Sciences
JF - Proceedings of the Japan Academy Series A: Mathematical Sciences
IS - 10
ER -