加乘数论与Ramanujan c(n)函数研究

加法数论和乘法数论是数论学科的两个重要分支。从2013年开始，申请人把加法和乘法数论结合起来考虑，可谓加乘数论或加乘方程（被德国同行称为阴阳方程）。例如华林问题、完美数问题等，费马大定理也可拓广为x+y=z，xyz=w^n在n和gcd（x,y,z）为奇素数时无正整数解。不同于费马大定理及已有的两个推广，新猜想在abc猜想假设下仍然坚挺。除了继续上述相关问题研究外，我们将利用加乘方程思想、椭圆曲线的群结构和挠点理论以及计算机辅助计算，进一步研究埃及分数、表整数为平方和问题拓广以及abcd方程等问题。一个世纪前，拉马努金在自守型意义下定义了函数τ(n)和c（n)，它们在怀尔斯有关费马大定理证明中起关键作用。关于τ(n)，他提出L级数乘积、绝对值上界和同余式三个猜想，其证明推动了20世纪数论和代数几何的发展。我们拟研究c(n)的相关性质并与加乘方程相联系，这些问题富有原创性且对数论研究很有意义。

Additive number theory and multiplicative number theory are two important branches of number theory. Since 2013, the applicant has considered the addition and multiplication equation together. We call it the addition-multiplication number theory or additive-multiplication equation (Yin and Yang equation according to a German numbe theorist). For example, Waring problem and Goldbach conjecture. FLT.can also be fully generalized. When n≥3, x + y = z, x y z = w ^ n has no positive integer solution under condition n and gcd (x, y, z) are odd primes. Unlike FLT and its other generalizations, this conjecture remains open under the abc conjecture. In addition to continuing study on above-mentioned problems, we will use the idea of addition-multiplication equation and the theory of elliptic curves to further study such problems as representing integer by square sum, ABCD equation, and Egyptian fraction. A century ago, Indian mathematician Ramanujan proposed two new functions τ(n)and c (n), which are related with the automorphic form, they play the key role in Wiles' proof of Fermat's last theorem. As for τ(n), he put forward three conjectures on L series, inequality and congruence. The proofs helps to promote the development of number theory and algebraic geometry in the 20th century. We intend to study the properties of c(n), including the congruence. The above problems and study will be very meaningful and original.