The new function of congruent number and its application in nine aspects

• 1、College of Information Enginnering,Hunan University
• 2、Department of Statictics and Financial Egninering,Ningbo university
• 3、 Beijing haitianqidian technical service co. LTD

Abstract：Abstract This paper uses a new function of the congruent number first proposed by the authors,Flexible application in the following seven aspects, achieved good results: 1.With this new function, we prove three new congruent negation theory of authors more concisely. 2.It is proved that A=1164714696873705 is not the congruent number.Much faster than the Tunnell theorem in this particular case. 3.Under the guidance ofThe new function of congruent number and its application in nine aspects the new function Z(A), the solution of congruent number 157 is obtained. Don Zagier does not make the algorithm public, and we are publishing it for the first time 4.Using this new function, the solution of the 127 is calculated for the first time, which is larger than the solution of 157. 5.Using this new function, the solution of the congruent number 822 is calculated for the first time, and many congruent number A can be calculated(which type is 112N) 6.Using this new function, we can calculate some congruences A (which type is N121) 7.By using this new function, combined with the sequential search method and the elliptic curve intersection method, it is proved that the number of solutions of the congruent number is 2,4,8,16,32, and there are infinitely many solutions under each type. 8.The fermat-Guanxungui theorem is developed and for the first time two new forms of necessary and sufficient conditions for the congruent number are found. 9.With this new function, we prove very succinctly the two negatives of the congruent number in historyIn this paper

Keywords： congruent number,elliptic curve,number theory,group