The positive integral points on the elliptic curve 𝒚𝟐=𝟕𝒑𝒙(𝒙𝟐+𝟖)

Abstract

The integral point of elliptic curve is a very important problem in both elementary number theory and analytic number theory. In recent years, scholars have paid great attention to solving the problem of positive integer points on elliptic curve 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), where 𝑘,𝑎,𝑏,𝑐 are integers. As a special case of 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), when 𝑎 = 1,𝑏 = 0,𝑐 = 22𝑡−1, it turns into 𝑦2 = 𝑘𝑥(𝑥2+22𝑡−1), which is a very important case. However ,at present, there are only a few conclusions on it, and the conclusions mainly concentrated on the case of 𝑡 = 1,2,3,4. The case of 𝑡 = 1, main conclusions reference [1] to [7]. The case of 𝑡 = 2, main conclusions reference [8]. The case of 𝑡 = 3, main conclusions reference [9] to [11]. The case of 𝑡 = 4, main conclusions reference [12] and [13]. Up to now, there is no relevant result on the case of 𝑘 = 7𝑝 when 𝑡 = 2, here the elliptic curve is 𝑦2 = 7𝑝(𝑥2 + 8), this paper mainly discusses the positive integral points of it. And we obtained the conclusion of the positive integral points on the elliptic curve 𝑦2 = 7𝑝(𝑥2 + 8). By using congruence, Legendre symbol and other elementary methods, it is proved that the elliptic curve in the title has at most one integer point when 𝑝 ≡ 5,7(𝑚𝑜𝑑8).