New {function, criterion, theorem, conjecture} for discriminant of congruence

[Zcyyu]

Originally Posted by Zcyyu

New function, new criterion, new theorem and new conjecture for discriminant of congruence
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 congruence discrimination is proposed,namely,the upper bound estimation funtion of the number of solutions of the congruence.
If the value of this function is 1,the posttive integer A is noncongruent,this is the new N-1 criterion.Using this guideline, to history all 3 congruence negative theorms,a very simple elementary proof is given;and proved the author's newly discovered three modular 8-type congruence negative theorems;
At the same time,eight new congruences negation theorems(not model 8 types) are proved,by the way,we derive 1511 infinite sequences of noncongruent
If A is a congruence number,the value of this function is integer power of 2,that is,there are at least two types of solutions for congruence,one of them must be Fermat type,and each type has an infinite number of solutions.
Using this upper bound function, some new theorems for the number of congruence solutions are derived: the exact value of the solution is 2, 4, 8, 16, 32;
On this basis, a new conjecture is put forward: If the type of the congruence solution is used as an element, then these elements form a group,and the order of this group is integer power of 2 ,for any congruence, the conversion relationship between the types is fixed.

Keyword：congruence number，Elliptic Curves，Number Theory,Group

Mod note: This post has been moved here from a new thread. Please don't make new threads with similar content to existing threads.

I've tried it out and can download the full PDF correctly

[Zcyyu]

Originally Posted by R.D. Silverman

It is the responsibility of the author to make the paper readable. Your knowledge
of English is far superior to my knowledge of Mandarin. However, the paper is
poorly written. May I suggest that you re-write the paper with the help of someone
with a better knowledge of English? You should also read the Halmos' book: How
to Write Mathematics.

At the beginning, we used the translation software,
but the quality was a little bad.
I'm going to revise it carefully

thanks !

[Zcyyu]

Originally Posted by Uncwilly

Can you provide a link to the paper or a pdf? English preferred.

PDF full text has been provided, thank you!

[Uncwilly]

Originally Posted by Zcyyu

PDF full text has been provided, thank you!

You have said that several times. Stop reposting the same message.

[Zcyyu]

Originally Posted by Zcyyu

Now we have provided the full PDF in English.

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=(s-t,t,s,s+t)=(1,1,157,1)
Using our preliminary theorem 3: the secondary solution is:
S=50356938758080675904478428415148993121355253942510969278703974330010718396658421418332558705681
T=49881830544284518392188087385168282973225882430995442036763690105298004209564650629551703029200
S-T= 〖21796977171070247104112455266586147721935979809〗^2
T=157*〖17824664537857719176051070357934327140032961660〗^2
S= 〖224403517704336969924557513090674863160948472041〗^2
S+T=〖316605068345983991287469841722668300352741098609〗^2
Fermat's type=(S-T,T,S,S+T)=(1,157,1,1)
The type group of solutions is 2-order, the relation matrix is as follows:
α F
α F α
F α F

[Zcyyu]

Originally Posted by xilman

If a man says something and there is no woman around to hear it, is he still wrong?

Three definitions of the congruent number
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^3-A^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=(s-t)ts(s+t) Is the area of the right triangle,a=s*s-t*t (odd side), b=2st(even side),c=s*s+t*t(hypotenuse)
where m is the maximum square factor of st(s*s-t*t),three squares are:x-A , x ,x+A,
here x=c*c/4/m/m
One solution to an elliptic curve is: x=c*c/4/m/m, y=c|a-b|(a+b)/8/m/m/m
We also define (s-t, 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^3-6^2x , 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 2-order 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)

[Zcyyu]

The group of type of solutions is 2-order group , the relation matrix is of type as follows:
.... α F
α F α
F α F

[Zcyyu]

Originally Posted by Zcyyu

The group of type of solutions is 2-order group , the relation matrix is of type as follows:
.... α F
α F α
F α F

Example: A=1164714696873705=3*5*17*41*73*89*137*257*487
∵Z(A)=1, Correctly judge A as a non-congruent number , Less than 3 minutes. Unfortunately, our method is only valid for about 50% the non-congruent 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 non-congruent 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 non-congruent number.After analysis and testing of the algorithm complexity, 10 years is not enough.

[Zcyyu]

ABstruct
In this paper, we present an N-1 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.
(non-module 8),by the way,we discover and publish 1511 non-congruent Number sequences.

Keyword：Congruent Number,Elliptic Curves,Number Theory

[paulunderwood]

"ABstruct" again

How do you feel today?

[Zcyyu]

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 non-congruent number in history.
3)Using new function,three new theorems of non-congruent 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