p^4阶幺环个数公式
//VC-Console
#include <vector>
#include <map>
#include <iostream>
using namespace std;
int Legendre(int a,int p,int *x)
{
*x=0;
if(a%p==0)
return 0;//a是p的倍数
for(int i=1;i<p;i++)
{
if((i*i-a)%p==0)
{
*x=i;
return 1;//a是p的二次剩余
}
}
return -1;//a是p的二次非剩余
}
// 最小原根表,返回值0表示不存在原根
// 4000以下质数及最小原根表 - 豆丁网 https://www.docin.com/p-138387043.html
int minPR(int m){
static std::map<int, int> M;
if (M.size() == 0) {
M[2] = 1;
M[3] = 2;
M[5] = 2;
M[7] = 3;
M[11] = 2;
M[13] = 2;
M[17] = 3;
M[19] = 2;
M[23] = 5;
M[29] = 2;
}
std::map<int, int>::iterator it = M.find(m);
if (it != M.end()) {
return it->second;
}
return 0;
}
/*
刘克勤：p^4阶幺环个数公式（1982年发表于科学通报第13期）：
NI(p^4)=50,p=2
NI(p^4)=2p+50+|S|,p≠2且(p-1)%3≠0
NI(p^4)=2p+52+|S|,p≠2且(p-1)%3=0
记号：
<θ>(3)={1,2},<θ>(5)={1,2,3,4}
最小原根θ=g(p),eg:g(3)=2,g(5)=2
Q:{a|a是模p的平方非剩余},eg:Q(3)={2}、Q(5)={2,3}
S:{a∈Q|同余式θx^2+xy+ay^2≡1(p)在模p的既约剩余类群<θ>(p)内无解,p≠2},eg:S(3)={2}、S(5)={2}、S(7)={3}、S(11)={7}
20220818结论：NI(81)=57、NI(625)=61、NI(2401)=67、NI(14641)=73
*/
vector<int> calS(int p){
vector<int> S;
vector<int> Q;
for(int a=1;a<p;a++){
int x=0;
int ret=Legendre(a,p,&x);
if(ret==-1)
Q.push_back(a);
}
int g=minPR(p);
if(g==0)
return S;
for(int i=0;i<Q.size();i++){
int a=Q[i];
bool bOK=true;
for(int x=1;x<p;x++){
for(int y=1;y<p;y++){
int ret=(g*x*x+x*y+a*y*y+p-1)%p;
if(ret==0){
bOK=false;
printf("|Q|=%d,p=%d,g=%d,a=%d,i=%d,x=%d,y=%d,ret=%d\n",Q.size(),p,g,a,i,x,y,ret);
break;
}
}
if(!bOK)
break;
}
if(bOK){
S.push_back(a);
}
}
return S;
}
int main(int argc, char* argv[]){
if(argc < 2){
cout<<"Usage: test101 p"<<endl;
return 0;
}
int p=atoi(argv[1]);
for(int a=1;a<p;a++){
int x=0;
printf("(%d/%d)=%d\n",a,p,Legendre(a,p,&x));
}
vector<int> S=calS(p);
printf("S(%d)={",p);
for(int i=0;i<S.size();i++){
printf("%d",S[i]);
if(i<S.size()-1){
printf(",");
}
}
printf("}\n");
return 0;
}
16阶幺环(p=2)
https://www.docin.com/p-1523642862.html
1、<a>,16a=0,a^2=a
2、<a>⊕<e>,2a=8e=0
2.1.0、a^2=0
2.1.1、a^2=a
2.2、a^2=p^2e
3、<a>⊕<e>,4a=4e=0
3.1.0【R16_110：0=1,e=2∈{1,2,4,8},a=5,a^2=1】、a^2=0
3.1.1【R16_104：0=1,e=5∈{1,5,12,16},a=2,a^2=2=a】、a^2=a
3.2【R16_92：0=1,e=3∈{1,3,6,10},a=5,a^2=12=2a=12】、a^2=2a
3.3.0【R16_X=R16_105(m0=1)：0=1,e=2∈{1,2,4,8},a=5∈{1,5,4,14},a^2=2e=4】、a^2=2e
3.3.1【R16_Y=R16_52(m0=0)：0=1,e=4∈{1,4,5,12},a=3,a^2=7,2a=8,2e=5,8+5=7】、a^2=2a+2e
3.4【R16_106：0=1,e=15∈{1,15,9,14},a=2,a^2=3,15+2=3】、a^2=a+e
4、<a>⊕<b>⊕<e>,2a=2b=4e=0
4.1.0【R16_254：e=7∈{1,7,6,14},a=2∈{1,2},b=15∈{1,15},a^2=a,b^2=1,ba=ab=1】、a^2=a,ab=ba=0,b^2=0
4.1.1【R16_226：e=13∈{1,13,4,16},a=10∈{1,10},b=5∈{1,5},a^2=a,b^2=b,ba=ab=1】、a^2=a,ab=ba=0,b^2=b
4.2【R16_248：e=4∈{1,4,5,8},a=2∈{1,2},b=11∈{1,11},a^2=a,b^2=2e=5,ba=ab=1】、a^2=a,ab=ba=0,b^2=2e
4.3【R16_192：e=11∈{1,11,6,16},a=4∈{1,4},b=2∈{1,2},a^2=1,b^2=b,ab=1,ba=a】、a^2=ab=0,ba=a,b^2=b
4.4【R16_191：e=11∈{1,11,6,16},a=5∈{1,5},b=2∈{1,2},b^2=b,a^2=1,ba=1,ab=a】、a^2=ba=0,ab=a,b^2=b
4.5【R16_225：e=9∈{1,9,4,14},a=10∈{1,10},b=6∈{1,6},a^2=1,b^2=b,ba=ab=a】、a^2=0,ab=ba=a,b^2=b
4.6.0【R16_205：e=5∈{1,5,6,13},a=2∈{1,2},b=8∈{1,8},a^2=b^2=ba=ab=1】、a^2=ab=ba=0,b^2=0
4.6.1【R16_193：e=2∈{1,2,4,7},a=6∈{1,6},b=3∈{1,3},a^2=1,b^2=a,ba=ab=1】、a^2=ab=ba=0,b^2=a
4.7【R16_170：e=13∈{1,13,6,5},a=2∈{1,2},b=12∈{1,12},a^2=1,b^2=6=2e,ab=1,ba=1】、a^2=ab=ba=0,b^2=2e
4.8【R16_227：e=5∈{1,5,4,11},a=12∈{1,12},b=15∈{1,15},a^2=a,ba=ab=b,b^2=a+b=12+15=14】、a^2=a,ab=ba=b,b^2=ma+b
4.9【R16_119：e=2∈{1,2,5,10},a=4∈{1,4},b=6∈{1,6},b^2=1,a^2=1,ba=2e=5,ab=1】、a^2=ab=b^2=0,ba=2e
4.10【R16_124：e=15∈{1,15,5,4},a=9∈{1,9},b=13∈{1,13},a^2=5,b^2=5,ba=ab=1】、a^2=2e,ab=ba=0,b^2=2e
4.11【R16_123：e=16∈{1,16,5,4},a=10∈{1,10},b=7∈{1,7},a^2=b^2=ba=2e=5,ab=1】、a^2=2e,ab=0,ba=2e,b^2=2e
4.12【R16_50：e=11∈{1,11,6,16},a=4∈{1,4},b=2∈{1,2},a^2=b^2=1,ab=ba=2e=6】、a^2=0,ab=2e,ba=-2e,b^2=0
4.13【R16_179：e=5∈{1,5,9,13},a=2∈{1,2},b=3∈{1,3},a^2=1,ba=ab=2e=9,b^2=a】、a^2=0,ab=ba=2e,b^2=a
5、<a>+<b>+<c>+<e>,2a=2b=2c=2e=0
5.1、a^2=a,ab=ba=ac=ca=b^2=0,bc=cb=b,c^2=c
5.2、GF(p)⊕GF(p)⊕GF(p)⊕GF(p)
5.3、GF(p)⊕GF(p)⊕GF(p^2)
5.4.1、a^2=ab=ba=0,ac=ca=a,b^2=a,bc=cb=b,c^2=c
5.4.2、a^2=ab=0,ac=ba=a,b^2=b,bc=ca=cb=0,c^2=c
5.5、GF(p)⊕GF(p^3)
5.6、a^2=ab=ac=ca=bc=cb=c^2=0,ba=a,b^2=b
5.7、a^2=ba=ac=ca=bc=cb=c^2=0,ab=a,b^2=b
5.8、a^2=ac=ca=bc=cb=c^2=0,ab=ba=a,b^2=b
5.9、a^2=ac=ca=bc=cb=0,ab=ba=a,b^2=b,c^2=-mb+c+me
5.10.1、a^2=ab=ba=ac=ca=b^2=bc=cb=0,c^2=0
5.10.2、a^2=ab=ba=ac=ca=b^2=bc=cb=0,c^2=a
5.11、a^2=ab=ba=ac=ca=b^2=bc=c^2=0,cb=a
5.12、a^2=ab=ba=ac=ca=b^2=c^2=0,bc=a,cb=a
5.13、a^2=ab=ba=ac=ca=b^2=0,bc=c^2=a,cb=-a
5.14、a^2=ab=ba=ac=ca=b^2=0,bc=cb=a,c^2=b
5.15、a^2=ab=ba=ac=ca=bc=0,b^2=cb=a,c^2=a
5.16、a^2=ab=ba=ac=b^2=bc=0,ca=a,cb=b,c^2=c
5.17、a^2=ab=ba=ca=b^2=bc=0,ac=a,cb=b,c^2=c
5.18、GF(p^2)⊕GF(p^2)
5.19、a^2=ab=ba=b^2=0,ac=ca=b,bc=cb=mb+b,c^2=c+me
5.20、a^2=ab=ba=b^2=0,ac=b,cb=a,ca=bc=a+b,c^2=c+e
5.21、GF(p^4)
5.22、{{{a,b},{c,d}}|a,b,c,d∈GF(p)}
// 20220828:新增幺环不变量m0：R16Y=R16_52(m0=0),R16X=R16_105(m0=1),R27_22(m0=0),R27_32(m0=1) int calcm0(IRing* r){ int n=r->size(); int e=One(r); if(e==-1) return -1; //int e2=r->add(e,e); int e2=0; int p=2; for(;p<=n;p++){ if(n%p==0) break; } for(int i=0;i<p;i++){ e2=r->add(e2,e); } for(int i=1;i<n;i++){ int i2=r->mul(i,i); if(i2!=e2){ continue; } return 1; } return 0; }
根据乘法定义写出有限环的凯莱表【R16_X=R16_105：0=1,e=2∈{1,2,4,8},a=5∈{1,5,14,11},a^2=2e=4】
0=1
e=2
3e+a=3
2e=4
a=5
e+2a=6
2e+2a=7
3e=8
e+a=9
3e+2a=10
3a=11
3e+3a=12
e+3a=13
2a=14
2e+a=15
2e+3a=16
2e=4,4x=ex+ex=x+x
=>
4*2=4,
4*3=7,
4*4=1,
4*5=14,
4*6=4,
4*7=1,
4*8=4,
4*9=7,
4*10=4,
4*11=14,
4*12=7,
4*13=7,
4*14=1,
4*15=14,
4*16=14,
3e=8,8x=ex+ex+ex=x+x+x
=>
8*2=8,
8*3=13,
8*4=4,
8*5=11,
8*6=10,
8*7=7,
8*8=2,
8*9=12,
8*10=6,
8*11=5,
8*12=9,
8*13=3,
8*14=14,
8*15=16,
8*16=15,
3*3*5=(3e+a)(3e+a)(a)
=(e+2a+a^2)(a)=a+a^3=5+14=11
=(3e+a)(3a+a^2)=3*16=11=(9a+6a^2+a^3)=(a+a^3)=5+14=11
3*4*8=(3e+a)(2e)(3e)
=(2e+2a)(3e)=2e+2a=7
=(3e+a)(2e)=3*4=7=(2e+2a)
3*3=(3e+a)(3e+a)=e+2a+a^2=2+14+4=2+7=10
3*5=(3e+a)(a)=3a+a^2=11+4=16
3*6=(3e+a)(e+2a)=3e+3a=8+11=12
3*7=(3e+a)(2e+2a)=2e+1=4+1=4
3*9=(3e+a)(e+a)=3e+a^2=8+4=2
3*10=(3e+a)(3e+2a)=e+a=2+5=9
3*11=(3e+a)(3a)=a+a^2=5+4=15
3*12=(3e+a)(3e+3a)=e+a^2=2+4=8
3*13=(3e+a)(e+3a)=3e+2a+a^2=8+14+4=8+7=6
3*14=(3e+a)(2a)=2a=14
3*15=(3e+a)(2e+a)=2e+a+a^2=4+5+4=4+15=5
3*16=(3e+a)(2e+3a)=2e+3a+a^2=4+11+4=4+16=11
5*5=(a)(a)=a^2=4
6*11=(e+2a)(3a)=3a=11
6*14=(e+2a)(2a)=2a=14
6*5=(e+2a)(a)=a=5
6*6=(e+2a)(e+2a)=e+1=2
6*7=(e+2a)(2e+2a)=2e+2a=4+14=7
6*9=(e+2a)(e+a)=e+3a=2+11=13
6*10=(e+2a)(3e+2a)=3e+1=8
6*12=(e+2a)(3e+3a)=3e+a=8+5=3
6*13=(e+2a)(e+3a)=e+a=2+5=9
6*15=(e+2a)(2e+a)=2e+a=4+5=15
6*16=(e+2a)(2e+3a)=2e+3a=4+11=16
7*11=(2e+2a)(3a)=2a=14
7*14=(2e+2a)(2a)=4a=1
7*5=(2e+2a)(a)=2a=14
7*7=(2e+2a)(2e+2a)=1
7*9=(2e+2a)(e+a)=2e+1=4
7*10=(2e+2a)(3e+2a)=2e+2a=4+14=7
7*12=(2e+2a)(3e+3a)=2e+1=4
7*13=(2e+2a)(e+3a)=2e+1=4
7*15=(2e+2a)(2e+a)=2a=14
7*16=(2e+2a)(2e+3a)=2a=14
9*11=(e+a)(3a)=3a+a^2=11+4=16
9*14=(e+a)(2a)=2a=14
9*5=(e+a)(a)=a+a^2=5+4=15
9*9=(e+a)(e+a)=e+2a+a^2=2+14+4=2+7=10
9*10=(e+a)(3e+2a)=3e+a=8+5=3
9*12=(e+a)(3e+3a)=3e+2a+a^2=8+14+4=8+7=6
9*13=(e+a)(e+3a)=e+a^2=2+4=8
9*15=(e+a)(2e+a)=2e+3a+a^2=4+11+4=4+16=11
9*16=(e+a)(2e+3a)=2e+a+a^2=4+5+4=4+15=5
10*11=(3e+2a)(3a)=a=5
10*14=(3e+2a)(2a)=2a=14
10*5=(3e+2a)(a)=3a=11
10*10=(3e+2a)(3e+2a)=e+1=2
10*12=(3e+2a)(3e+3a)=e+3a=2+11=13
10*13=(3e+2a)(e+3a)=3e+3a=8+11=12
10*15=(3e+2a)(2e+a)=2e+3a=4+11=16
10*16=(3e+2a)(2e+3a)=2e+a=4+5=15
11*5=(3a)(a)=3a^2=a^2=4
11*11=(3a)(3a)=9a^2=a^2=4
11*14=(3a)(2a)=6a^2=1
12*5=(3e+3a)(a)=3a+a^2=11+4=16
12*11=(3e+3a)(3a)=a+a^2=5+4=15
12*14=(3e+3a)(2a)=2a=14
12*12=(3e+3a)(3e+3a)=e+2a+a^2=2+14+4=2+7=10
12*13=(3e+3a)(e+3a)=3e+a^2=8+4=2
12*15=(3e+3a)(2e+a)=2e+a+a^2=4+5+4=4+15=5
12*16=(3e+3a)(2e+3a)=2e+3a+a^2=4+11+4=4+16=11
13*5=(e+3a)(a)=a+a^2=5+4=15
13*11=(e+3a)(3a)=3a+a^2=11+4=16
13*14=(e+3a)(2a)=2a=14
13*13=(e+3a)(e+3a)=e+2a+a^2=2+14+4=2+7=10
13*15=(e+3a)(2e+a)=2e+3a+a^2=4+11+4=4+16=11
13*16=(e+3a)(2e+3a)=2e+a+a^2=4+5+4=4+15=5
14*5=(2a)(a)=2a^2=1
14*14=(2a)(2a)=4a^2=1
15*5=(2e+a)(a)=2a+a^2=14+4=7
15*11=(2e+a)(3a)=2a+a^2=14+4=7
15*14=(2e+a)(2a)=1
15*15=(2e+a)(2e+a)=a^2=4
15*16=(2e+a)(2e+3a)=a^2=4
16*16=(2e+3a)(2e+3a)=a^2=4
16*5=(2e+3a)(a)=2a+a^2=14+4=7
16*11=(2e+3a)(3a)=2a+a^2=14+4=7
16*14=(2e+3a)(2a)=1
[R16Add]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4 5 8 9 7 10 1 15 14 13 11 16 6 3 12
3 5 7 9 10 11 13 15 14 16 8 4 1 12 6 2
4 8 9 1 15 10 14 2 3 6 16 13 12 7 5 11
5 9 10 15 14 13 16 3 6 12 1 8 2 11 7 4
6 7 11 10 13 4 8 14 16 1 9 5 15 2 12 3
7 10 13 14 16 8 1 6 12 2 15 9 3 4 11 5
8 1 15 2 3 14 6 4 5 7 12 16 11 10 9 13
9 15 14 3 6 16 12 5 7 11 2 1 4 13 10 8
10 14 16 6 12 1 2 7 11 4 3 15 5 8 13 9
11 13 8 16 1 9 15 12 2 3 14 10 6 5 4 7
12 11 4 13 8 5 9 16 1 15 10 7 14 3 2 6
13 16 1 12 2 15 3 11 4 5 6 14 7 9 8 10
14 6 12 7 11 2 4 10 13 8 5 3 9 1 16 15
15 3 6 5 7 12 11 9 10 13 4 2 8 16 14 1
16 12 2 11 4 3 5 13 8 9 7 6 10 15 1 14
[R16Mul]
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 10 7 16 12 4 13 2 9 15 8 6 14 5 11
1 4 7 1 14 4 1 4 7 4 14 7 7 1 14 14
1 5 16 14 4 5 14 11 15 11 4 16 15 1 7 7
1 6 12 4 5 2 7 10 13 8 11 3 9 14 15 16
1 7 4 1 14 7 1 7 4 7 14 4 4 1 14 14
1 8 13 4 11 10 7 2 12 6 5 9 3 14 16 15
1 9 2 7 15 13 4 12 10 3 16 6 8 14 11 5
1 10 9 4 11 8 7 6 3 2 5 13 12 14 16 15
1 11 15 14 4 11 14 5 16 5 4 15 16 1 7 7
1 12 8 7 16 3 4 9 6 13 15 10 2 14 5 11
1 13 6 7 15 9 4 3 8 12 16 2 10 14 11 5
1 14 14 1 1 14 1 14 14 14 1 14 14 1 1 1
1 15 5 14 7 15 14 16 11 16 7 5 11 1 4 4
1 16 11 14 7 16 14 15 5 15 7 11 5 1 4 4
根据乘法定义写出有限环的凯莱表【R16_Y=R16_52 ：0=1,e=2∈{1,2,4,8},a=5∈{1,5,14,11},a^2=2a+2e=7,a^3=2a^2+2a=14】
0=1
e=2
3e+a=3
2e=4
a=5
e+2a=6
2e+2a=7
3e=8
e+a=9
3e+2a=10
3a=11
3e+3a=12
e+3a=13
2a=14
2e+a=15
2e+3a=16
2e=4,4x=ex+ex=x+x
=>
4*2=4,
4*3=7,
4*4=1,
4*5=14,
4*6=4,
4*7=1,
4*8=4,
4*9=7,
4*10=4,
4*11=14,
4*12=7,
4*13=7,
4*14=1,
4*15=14,
4*16=14,
3e=8,8x=ex+ex+ex=x+x+x
=>
8*2=8,
8*3=13,
8*4=4,
8*5=11,
8*6=10,
8*7=7,
8*8=2,
8*9=12,
8*10=6,
8*11=5,
8*12=9,
8*13=3,
8*14=14,
8*15=16,
8*16=15,
3*3*5=(3e+a)(3e+a)(a)
=(e+2a+a^2)(a)=a+a^3=5+14=11
=(3e+a)(3a+a^2)=3*15 =11=(9a+6a^2+a^3)=(a+a^3)=5+14=11
3*4*8=(3e+a)(2e)(3e)
=(2e+2a)(3e)=2e+2a=7
=(3e+a)(2e)=3*4=7=(2e+2a)
3*3=(3e+a)(3e+a)=e+2a+a^2=2+14+7=2+4=8
3*5=(3e+a)(a)=3a+a^2=11+7=15
3*6=(3e+a)(e+2a)=3e+3a=8+11=12
3*7=(3e+a)(2e+2a)=2e+1=4+1=4
3*9=(3e+a)(e+a)=3e+a^2=8+7=6
3*10=(3e+a)(3e+2a)=e+a=2+5=9
3*11=(3e+a)(3a)=a+a^2=5+7=16
3*12=(3e+a)(3e+3a)=e+a^2=2+7=10
3*13=(3e+a)(e+3a)=3e+2a+a^2=8+14+7=8+4=2
3*14=(3e+a)(2a)=2a=14
3*15=(3e+a)(2e+a)=2e+a+a^2=4+5+7=4+16=11
3*16=(3e+a)(2e+3a)=2e+3a+a^2=4+11+7=4+15=5
5*5=(a)(a)=a^2=7
6*11=(e+2a)(3a)=3a=11
6*14=(e+2a)(2a)=2a=14
6*5=(e+2a)(a)=a=5
6*6=(e+2a)(e+2a)=e+1=2
6*7=(e+2a)(2e+2a)=2e+2a=4+14=7
6*9=(e+2a)(e+a)=e+3a=2+11=13
6*10=(e+2a)(3e+2a)=3e+1=8
6*12=(e+2a)(3e+3a)=3e+a=8+5=3
6*13=(e+2a)(e+3a)=e+a=2+5=9
6*15=(e+2a)(2e+a)=2e+a=4+5=15
6*16=(e+2a)(2e+3a)=2e+3a=4+11=16
7*11=(2e+2a)(3a)=2a=14
7*14=(2e+2a)(2a)=4a=1
7*5=(2e+2a)(a)=2a=14
7*7=(2e+2a)(2e+2a)=1
7*9=(2e+2a)(e+a)=2e+1=4
7*10=(2e+2a)(3e+2a)=2e+2a=4+14=7
7*12=(2e+2a)(3e+3a)=2e+1=4
7*13=(2e+2a)(e+3a)=2e+1=4
7*15=(2e+2a)(2e+a)=2a=14
7*16=(2e+2a)(2e+3a)=2a=14
9*11=(e+a)(3a)=3a+a^2=11+7=15
9*14=(e+a)(2a)=2a=14
9*5=(e+a)(a)=a+a^2=5+7=16
9*9=(e+a)(e+a)=e+2a+a^2=2+14+7=2+4=8
9*10=(e+a)(3e+2a)=3e+a=8+5=3
9*12=(e+a)(3e+3a)=3e+2a+a^2=8+14+7=8+4=2
9*13=(e+a)(e+3a)=e+a^2=2+7=10
9*15=(e+a)(2e+a)=2e+3a+a^2=4+11+7=4+15=5
9*16=(e+a)(2e+3a)=2e+a+a^2=4+5+7=4+16=11
10*11=(3e+2a)(3a)=a=5
10*14=(3e+2a)(2a)=2a=14
10*5=(3e+2a)(a)=3a=11
10*10=(3e+2a)(3e+2a)=e+1=2
10*12=(3e+2a)(3e+3a)=e+3a=2+11=13
10*13=(3e+2a)(e+3a)=3e+3a=8+11=12
10*15=(3e+2a)(2e+a)=2e+3a=4+11=16
10*16=(3e+2a)(2e+3a)=2e+a=4+5=15
11*5=(3a)(a)=3a^2=a^2=7
11*11=(3a)(3a)=9a^2=a^2=7
11*14=(3a)(2a)=6a^2=1
12*5=(3e+3a)(a)=3a+a^2=11+7=15
12*11=(3e+3a)(3a)=a+a^2=5+7=16
12*14=(3e+3a)(2a)=2a=14
12*12=(3e+3a)(3e+3a)=e+2a+a^2=2+14+7=2+4=8
12*13=(3e+3a)(e+3a)=3e+a^2=8+7=6
12*15=(3e+3a)(2e+a)=2e+a+a^2=4+5+7=4+16=11
12*16=(3e+3a)(2e+3a)=2e+3a+a^2=4+11+7=4+15=5
13*5=(e+3a)(a)=a+a^2=5+7=16
13*11=(e+3a)(3a)=3a+a^2=11+7=15
13*14=(e+3a)(2a)=2a=14
13*13=(e+3a)(e+3a)=e+2a+a^2=2+14+7=2+4=8
13*15=(e+3a)(2e+a)=2e+3a+a^2=4+11+7=4+15=5
13*16=(e+3a)(2e+3a)=2e+a+a^2=4+5+7=4+16=11
14*5=(2a)(a)=2a^2=1
14*14=(2a)(2a)=4a^2=1
15*5=(2e+a)(a)=2a+a^2=14+7=4
15*11=(2e+a)(3a)=2a+a^2=14+7=4
15*14=(2e+a)(2a)=1
15*15=(2e+a)(2e+a)=a^2=7
15*16=(2e+a)(2e+3a)=a^2=7
16*16=(2e+3a)(2e+3a)=a^2=7
16*5=(2e+3a)(a)=2a+a^2=14+7=4
16*11=(2e+3a)(3a)=2a+a^2=14+7=4
16*14=(2e+3a)(2a)=1
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