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Solve Goldbach Conjecture with Chandra Matrix
Computation
用钱德拉矩阵解算哥德巴赫猜想 - 主讲人周密
By
Mi Zhou, Jun S. Huang, Qi Chen
The 3rd International Conference on Mathematical Science,
2016. 4. 16 Beijing China
About the authors
• Zhou Mi is student in Huaiyin Institute of Technology,graduated
from Suqian Economic and Trade Vocational College recently.
His current research interest is number theory, Mathematical
Olympiad mentoring.
• Jun Steed Huang, his doctor's degree in 1992 from a Joint Ph.D
program between Southeast University China and Concordia
University Canada. He worked at Bell Canada, Lockheed Martin
USA, Ottawa University. He is a Professor of Suqian College with
Jiangsu University. He has been invited as board level advisor for
a number of organizations from North American.
• Chen Qi , B.E. degree in Department of Telecommunications
Engineering form Jiangsu University , now studying for master at
Shanghai. His technical interest includes wireless sensor image
processing and network programming.
Agenda
1. BACKGROUND
2. INTRODUCTION
3. PROOF PRINCIPLE
4. MATLAB SIMULATION
5. CONCLUSION AND FUTURE WORK
1.BACKGROUND
Goldbach's original conjecture states "at
least it seems that every number that is
greater than 2 is the sum of three primes”
Today it stands as: “Every even integer
greater than 4 can be expressed as the
sum of two primes”
But remains unproven in a strictly
mathematical sense, despite considerable
effort from researchers all over the world.
德国俄国民间数学家律师 1690-1764
2. INTRODUCTION: METHODS
• There are a number of ways to approach this
conjecture , as follows• The circle method , the most authentic way but
hard to understand• The Chen’s Sieve , still not easy to get it• Probabilistic analysis , preferred more by
engineers• The matrix method , very intuitive with some
construction of relatively large numbers
3. PROOF PRINCIPLE
• Chandra matrix is a square sieve with the first row of the square sieve consists of the first element of 4, the
difference between next every two adjacent numbers is 3
if number N in the table then 2N+1
is not a prime number
if number N does not appear then
2N+1 is a prime number
印度孟加拉国美国半单李代数数学物理学家 1923-1983
• Based on above observations, we made a few
similar matrices accordingly
• if number N appear in the
• matrix then 2*N-(2x-1) is a
• not a prime number
• if number N does not appear
• then 2*N-(2x-1) must be a
• prime number
• Notice that minuend 2N is an even number, and
the subtrahend are all odd numbers. there are
four kinds of situations:
素数金字塔 ABCD 两两相间守恒
• In summary, any even number not less than 40
can be expressed as an odd composite number plus an odd composite number.
• So the N in D include all even numbers big than 20, equivalently the N in A also contains all the
numbers,
• case A: 2N- prime number = prime number then 2N can be expressed as the sum of two prime
numbers, so all the even numbers greater than
40 can be expressed as the sum of two prime
numbers.
4. MATLAB COMPUTATIONS
• The result of each case with matlab
computations is shown below :
• The figure shows the results of 2N=400.
A 与 D 相间
自对称守恒
B 与 C 相反
互对称守恒
• From which we can clearly see that
• The Case A and Case D are strictly symmetrical by itself, rooted from the interleaved symmetrical Chandra matrix itself,
• The Case B and Case C are symmetrical with
respect to each other rooted from the row
versus column lifting effect. • From Case A, we can see that the prime is
always mirrorly paired with some one else, or
itself if it sits right on the mirror at 45 degree’s
position.
5.CONCLUSION AND FUTURE WORK
• All the even numbers great than 40 can be
expressed as the sum of two prime numbers,
finding these primes is hard by using hand
calculation though. • This matrix based method is supported with the
Matlab program available on Matlab server ：
http://www.mathworks.com/matlabcentral/fileexchange/52258-goldbach-partition-with-chandra-sieve
• We can use it to fulfil the tasks of the public or
private key generation and distribution, with the variations of two prime partition algorithms.
