按：承蒙OMICS出版集团旗下的Journal of Applied & Computational Mathematics（《应用数学与计算数学杂志》）主编相邀，担任该刊编委(编辑部设在洛杉矶），这是博主为即将创刊的杂志撰写的发刊文章（每位编委都得写一篇）。其中有这样一句话，“计算机对于数论学家，其重要性不亚于望远镜对于天文学家。”

              More Comprehensive Computational Number Theory

Tianxin Cai， University of Zhejiang,  Department of Mathematics,  Hangzhou, 310027, China

    Computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is the  integer factorization, the great common divisor or the least common multiple. The largest known primes, which are usually Mersenne primes, and they were usually found by GIMPS, the Great Internet Mersenne Prime Search，etc..

    But in my opinion, the concepts of computational number theory could be more comprehensive, covering a wide range,  including all the study of number theory by computers.  For me, computers are so important for number theorists, just like telescopes for astronomers.

For example, recently, I found that  except 2, 5 and 11, every positive prime can be expressed as a sum of three positive integers a, b, c， the  product abc is a cube. For instance, 3 = 1+1+1，7 = 1+2+4，13 = 1+3+9 ，17 = 1+8+8， 19 = 4+6+9 and  1·1·1 = 1^3, 1·2·4 = 2^3,   1·3·9=3^3, 1·8·8 = 4^3, 1·8·8 = 4^3, 4·6·9= 6^3. 

    By using a computer, we have tested  that it is true for primes less  than  10000. So we make the above conjecture that Except for 1; 2; 4; 5; 8; 11; 16; 22; 32; 44; 88; 176, every positive integer can be expressed as a sum of three positive integers where the product of those integers is a cube.

    Similarly, we conjecture that Except for 1; 2; 3; 5; 6; 7; 11; 13; 14; 15; 17; 22; 23, every positive integer can be expressed as a sum of four positive integers where the product of those integers is a 4-th power.

    Open Access is a new way for us  to publish and read research papers, it’s easy and quick, especially in the present time of world  economy crisis. I believe that  all of the  mathematicians, particular in developing countries and small company will benefit for it, since they might not offer the expensive fees if journal subscriptions. Therefore, I recommend that the OMICS' Applied & Computational Mathematics Open Access policy (see http://www.omicsonline.org/OpenAccess.php for details) and the special features that the publisher provides (see http://www.omicsonline.org/special-features.php)

    Further readings

   1. Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, volume 1: Efficient Algorithms. MIT Press, 1996.

   2. Tianxin Cai and Deyi Chen, A new variant of the Hilbert-Waring  Problem, to appear in Mathematics of Computation.

   3. Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, 2001.

   4. Henri Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, 1993.

   5. Hans Riesel, Prime Numbers and Computer Methods for Factorization, second edition, Birkhäuser, 1994,

   6. Victor Shoup, A Computational Introduction to Number Theory and Algebra. Cambridge, 2005.