The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infini...The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.展开更多
The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross su...The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross subsidy. Based on the status of our sales price and cross subsidy, Ramsey pricing methods theoretically guide adjustment thoughts of sales price and solve the practical problems in our life.展开更多
Given a graph G and a positive integer k,define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow(all different colored)triangle or a monoch...Given a graph G and a positive integer k,define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow(all different colored)triangle or a monochromatic copy of G.In this paper,we obtain exact values of the Gallai–Ramsey numbers for the union of two stars in many cases and bounds in other cases.This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.展开更多
文摘The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.
文摘The Ramsey number is a foundational result in combinatorics. This article will introduce Ramsey number with the method of graph theory, and the Ramsey pricing theory is applied to the sales price and study of cross subsidy. Based on the status of our sales price and cross subsidy, Ramsey pricing methods theoretically guide adjustment thoughts of sales price and solve the practical problems in our life.
基金Supported by the National Science Foundation of China(Grant Nos.12061059 and 61763041)。
文摘Given a graph G and a positive integer k,define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow(all different colored)triangle or a monochromatic copy of G.In this paper,we obtain exact values of the Gallai–Ramsey numbers for the union of two stars in many cases and bounds in other cases.This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.
