两年以前，受abc猜想的形式启发，我们把整数的加法和乘法结合起来考虑，针对希尔伯特-华林问题，提出了自己的想法，经过尝试和努力，取得了很好的效果，且不失深度和难度。现在，我们想把这个思想应用于完美数问题、费尔马大定理、哥德巴赫问题、同余数等经典问题，提出一些新的问题和猜想，并解决或部分解决其中的若干问题。在使用工具上，除了初等数论和解析数论的基本方法以外，椭圆曲线理论也将起重要作用。随着费尔马大定理的证明，这个古老的数学概念又焕发出新的活力。而把加法和乘法结合起来（如同物理学家探究质子和中子之间的关系）以后，上述某些问题可归结为椭圆曲线的整点和有理点求解，包括abcd方程（定义见正文）在内的新问题也因此得以提出。如同一些数论大家所注意到的，考察某些丢番图方程的整数或有理数解会引导我们到数学的深处，本项目的选题新颖、深刻而有意义，成果的取得将帮助国内外同行拓宽和丰富数论领域的研究。

Two years ago, inspired by the form of abc-conjecture, we considered the addition and multiplication of integers together, raised a variant of Hilbert-Waring's problem. After great effort, we made beautiful progress without losing its depth and difficulty. Now, we want to apply this idea to perfect numbers, Fermat's last theorem, Goldbach conjecture, congruent numbers and other claasic number theory problems, meanwhile, we will raise several new problems and conjectures, solve or partly solve some of them . As for the tool and method, we will use basic number theory and analytic number theory, especially the properties and theory of elliptic curves. Since FLT was proved, the classic mathematical concept of elliptic curves is active and attractive again, number theoriests pay more and more attention to it. We find that the combination of addition and mulplication of integers （meaningful as for physicists exploring the relation of protons and neutrons）could transform some of the problems to the integral or rational solutions of the elliptic curves, and help to rasie new problems such as new congruenct problem and abcd-equation. As many of the great mathematicians noticed, the studying and investigation of the integral and rational solutions of some Diophantine equations will lead us to the deepness in mathematics. The problems and plans of this project are new, profound and meaningful, we believe that it will help us to wide and enrich the study of contemporary number theory.