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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  2  = ( 2 ++), where ,,, are integers. As a special case of  2  = ( 2 ++), when  = 1, = 0, = 2 2−1 , it turns into  2  = ( 2 +2 2−1 ), 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  = 1,2,3,4. The case of  = 1, main conclusions reference [1] to [7]. The case of  = 2, main conclusions reference [8]. The case of  = 3, main conclusions reference [9] to [11]. The case of  = 4, main conclusions reference [12] and [13]. Up to now, there is no relevant result on the case of  = 7 when  = 2, here the elliptic curve is  2  = 7( 2  + 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  2  = 7( 2  + 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  ≡ 5,7(8).

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

The positive integral points on the elliptic curve 𝒚^𝟐=𝟕𝒑𝒙(𝒙^𝟐+𝟖)