A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given h...A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given ho and B completely. Also, using a recently result of Ribet, Darmon and Merel, we proved that Erdos’ conjecture on combinatorial number is right.展开更多
This paper syggests the power vector and the vector power series. The vector power series is a 'Combinatorial Vector Number' which is composed of a real number and a certain vector. Combinatorial rector variab...This paper syggests the power vector and the vector power series. The vector power series is a 'Combinatorial Vector Number' which is composed of a real number and a certain vector. Combinatorial rector variable and its functions have an important meaning. They have the fundamental operations of arithemetic too. From the theoretical analysis of functions for a combinatorial vector variable we would would know its function has the derivatives. the necessary and sufficient conditions. These conditions make up the charaders of 'Hyperbolic equation' and its integralions.展开更多
This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, ...This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.展开更多
基金Supported by the Natural Science Foundation of Heilongjiang Province
文摘A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given ho and B completely. Also, using a recently result of Ribet, Darmon and Merel, we proved that Erdos’ conjecture on combinatorial number is right.
文摘This paper syggests the power vector and the vector power series. The vector power series is a 'Combinatorial Vector Number' which is composed of a real number and a certain vector. Combinatorial rector variable and its functions have an important meaning. They have the fundamental operations of arithemetic too. From the theoretical analysis of functions for a combinatorial vector variable we would would know its function has the derivatives. the necessary and sufficient conditions. These conditions make up the charaders of 'Hyperbolic equation' and its integralions.
基金Supported by the National Natural Science Foundation of China (Grant No.11071183)
文摘This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.
