In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the ...

Let p ≡ 4, 7 mod 9 be a rational prime number such that 3 mod p is not a cube. In this paper, we prove the 3-part of | III( E p ) |⋅| III( E 3 p 2 ) | is as predicted by the Birch and Swinnerton-Dyer ...