20200704, 08:40  #23  
Jun 2020
23 Posts 
New function, new criterion, new theorem and new conjecture for discriminant of congruence number
Quote:


20200704, 11:21  #24  
Jun 2020
23 Posts 
New function, new criterion, new theorem and new conjecture for discriminant of congruent number
Quote:
but the quality was a little bad. I'm going to revise it carefully thanks ! 

20200708, 04:21  #25 
Jun 2020
23 Posts 
New {function, criterion, theorem, conjecture} for discriminant of congruent number

20200708, 06:19  #26 
6809 > 6502
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Aug 2003
101×103 Posts
3·5·613 Posts 

20200709, 08:40  #27 
Jun 2020
17_{16} Posts 
A new function and new discovery of the number of type of solution about the congruent number
For your reference：
The calculation of the congruent number is very difficult. Here is an example: The smallest solution of 157,provided by genius mathematician Don Zagier, but he didn't provide an algorithm. After my conversion,get s=443624018997429899709925, t=166136231668185267540804 Area of right triangle =157*〖8912332268928859588025535178967163570016480830〗^2 We define the type of this solution as α=(st,t,s,s+t)=(1,1,157,1) Using our preliminary theorem 3: the secondary solution is: S=50356938758080675904478428415148993121355253942510969278703974330010718396658421418332558705681 T=49881830544284518392188087385168282973225882430995442036763690105298004209564650629551703029200 ST= 〖21796977171070247104112455266586147721935979809〗^2 T=157*〖17824664537857719176051070357934327140032961660〗^2 S= 〖224403517704336969924557513090674863160948472041〗^2 S+T=〖316605068345983991287469841722668300352741098609〗^2 Fermat's type=(ST,T,S,S+T)=(1,157,1,1) The type group of solutions is 2order, the relation matrix is as follows: α F α F α F α F 
20200709, 22:42  #28  
Jun 2020
23 Posts 
Three definitions of the congruent number
Quote:
1) The oldest definition：Look for the squares of the three isometric differences, This tolerance is called the congruent number,. E.g: 1/4 ,25/4 ,49/4, The tolerance is 6, This 6 is called the congruent number. 2) The area of a rational right triangle： If the area of a right triangle with three sides of 3,4, and 5 is 6, the 6 is the congruent number.. 3) The modern definition: If the elliptic curve y^2=x^3A^2 x There is a solution for y≠0, then A is the congruent number. We use the parameter (s,t) to unify the above three definitions: let s,t be positive integers,one odd and one even,(s,t)=1,then Am^2=(st)ts(s+t) Is the area of the right triangle,a=s*st*t (odd side), b=2st(even side),c=s*s+t*t(hypotenuse) where m is the maximum square factor of st(s*st*t),three squares are:xA , x ,x+A, here x=c*c/4/m/m One solution to an elliptic curve is: x=c*c/4/m/m, y=cab(a+b)/8/m/m/m We also define (st, t,s,s+t) as the type of solution, Furthermore,the type of solutions form a group is derived. (our new developments in field of congruent number) E.g s=2,t=1 ,then the three squares are 1/4,25/4,49/4. Tolerance 6 is congruent number. If three sides of a right triangle are 3,4,5 then the area is 6.m is 1,this 6 is congruent number. The elliptic curve y^2=x^36^2x , there is a solution, (25/4,35/8).so this 6 is congruent number There are two types of solutions for 6, this is (1,1,2,3) and (1,6,1,1) The latter is called the Fermat type. The type group of 6 is 2order group. By our method, get a lot of new results for the congruent number, For details,see the full text PDF. (Probably in a paper dry goods is the most) Last fiddled with by Zcyyu on 20200709 at 22:47 Reason: Small changes 

20200710, 22:50  #29 
Jun 2020
23 Posts 
New {function, new criterion, new theorem and new conjecture} about congruent number
The group of type of solutions is 2order group , the relation matrix is of type as follows:
.... α F α F α F α F Last fiddled with by Uncwilly on 20200710 at 23:22 Reason: Deleted all unneeded self quote. 
20200711, 04:12  #30  
Jun 2020
23 Posts 
Example: A=1164714696873705=3*5*17*41*73*89*137*257*487
Quote:
∵Z(A)=1, Correctly judge A as a noncongruent number , Less than 3 minutes. Unfortunately, our method is only valid for about 50% the noncongruent number of 10 million the following(Correct judgment 1445438), but the new function still has room for development. As long as new methods are found outside of qij and M8, the world is looking forward to it The Tunnell method is effective for all noncongruent numbers (all of 2826325 effective), but it is too slow; using the Tunnell method, the same machine and the same algorithm language, prove that this A is a noncongruent number.After analysis and testing of the algorithm complexity, 10 years is not enough. 

20200712, 01:47  #31 
Jun 2020
23 Posts 
N1 criterion for discriminant,8 new theorems and 1511 sequences about congruent Number
ABstruct
In this paper, we present an N1 criterion for discriminating congruent Number, and use this criterion to prove that three theories of congruent Numbers in history. It is also proved that 3 new theorems of congruent Numbers found by the author(modele 8). At the same time,it is proved that 8 new theorems of congruent Numbers found by the author. (nonmodule 8),by the way,we discover and publish 1511 noncongruent Number sequences. Keyword：Congruent Number,Elliptic Curves,Number Theory 
20200712, 01:56  #32 
Sep 2002
Database er0rr
2^{2}·883 Posts 
"ABstruct" again
How do you feel today? 
20200712, 04:18  #33 
Jun 2020
23 Posts 
A new function and new discovery of the number of type of solution about the congruent number
A new function and new discovery of the number of type of solution about the congruent number Zhou CongYao College of Information Science and Engineering, Hunan University
Yu Wei Department of statistics & financial engineering,College of mathematics and statistics,ningbo university TangXiaoNing Beijing Haitian Start Technology Service Co., Ltd. ABstruct In this paper,a new function of congruent number is proposed,namely,the upper bound estimation funtion of the number of type of solutions of the A.Named Z(A),this A is square free positive integer. 1)If the value of this function is 1,the posttive integer A is noncongruent number. 2)By this new function,we prove very simply three test theorems of noncongruent number in history. 3)Using new function,three new theorems of noncongruent number of authors. 4)By using this new function, by nine new theromes of ours, it is proved that the type number of solution of many congruent number is 2 or 4 or 8 or 16 or 32,each type has an infinite number of solutions 5)On this basis, a new conjecture is put forward: If the type of solution of the congruent number is used as an element, then these elements form a group,and the order of this group is integer power of 2 ,for any congruent number, the relationship between the types is fixed. Keyword：congruent number,Elliptic Curves,Number Theory,Group Last fiddled with by Zcyyu on 20200712 at 04:21 Reason: According to the expert's suggestion, has made many modifications to the original text 
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