R32R48R64R128R256的编号

Posted 2022-12-14 华仔Ivan

有限环分类问题，是一项机械枯燥、计算量巨大的基础数学领域的计算工作，单个人、单台计算机短时间内很难完成，需要众多数学爱好者加入进来共同完成这项有意义的工作。
32阶环至少有18590种，48阶环有780种，64阶环至少有829826种。
结论来源：
The sequence continues a(32) = ? (>18590), a(33) = 4, 4, 4, 121, 2, 4, 4, 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, 4, 22, 2, 118, 4, 104, 4, 4, 2, 44, 2, 4, 22 = a(63), a(64) = ? (> 829826). - Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
gap> calT:=function(n) local L,m;L:=DivisorsInt(n);m:=Size(L);return m;end;;
gap> for n in [32,48,64,128,256] do Print("T(",n,")=",calT(n),"\\n");od;
T(32)=6
T(48)=10
T(64)=7
T(128)=8
T(256)=9
确定编号空间之前，先看下它们的加法群：
gap> n:=32;;m:=NumberSmallGroups(n);;for i in [1..m] do G:=SmallGroup(n,i);;if IsAbelian(G) then L:=List(Elements(G),Order);M:=DivisorsInt(n);Print("G",n,"_",i,"=",StructureDescription(G),",N0=");for k in M do Print(Size(Positions(L,k)),",");od;Print("\\n");fi;od;
G32_1=C32,N0=1,1,2,4,8,16,
G32_3=C8 x C4,N0=1,3,12,16,0,0,
G32_16=C16 x C2,N0=1,3,4,8,16,0,
G32_21=C4 x C4 x C2,N0=1,7,24,0,0,0,
G32_36=C8 x C2 x C2,N0=1,7,8,16,0,0,
G32_45=C4 x C2 x C2 x C2,N0=1,15,16,0,0,0,
G32_51=C2 x C2 x C2 x C2 x C2,N0=1,31,0,0,0,0,
gap> n:=48;;m:=NumberSmallGroups(n);;for i in [1..m] do G:=SmallGroup(n,i);;if IsAbelian(G) then L:=List(Elements(G),Order);M:=DivisorsInt(n);Print("G",n,"_",i,"=",StructureDescription(G),",N0=");for k in M do Print(Size(Positions(L,k)),",");od;Print("\\n");fi;od;
G48_2=C48,N0=1,1,2,2,2,4,4,8,8,16,
G48_20=C12 x C4,N0=1,3,2,12,6,0,24,0,0,0,
G48_23=C24 x C2,N0=1,3,2,4,6,8,8,0,16,0,
G48_44=C12 x C2 x C2,N0=1,7,2,8,14,0,16,0,0,0,
G48_52=C6 x C2 x C2 x C2,N0=1,15,2,0,30,0,0,0,0,0,
gap> n:=64;;m:=NumberSmallGroups(n);;for i in [1..m] do G:=SmallGroup(n,i);;if IsAbelian(G) then L:=List(Elements(G),Order);M:=DivisorsInt(n);Print("G",n,"_",i,"=",StructureDescription(G),",N0=");for k in M do Print(Size(Positions(L,k)),",");od;Print("\\n");fi;od;
G64_1=C64,N0=1,1,2,4,8,16,32,
G64_2=C8 x C8,N0=1,3,12,48,0,0,0,
G64_26=C16 x C4,N0=1,3,12,16,32,0,0,
G64_50=C32 x C2,N0=1,3,4,8,16,32,0,
G64_55=C4 x C4 x C4,N0=1,7,56,0,0,0,0,
G64_83=C8 x C4 x C2,N0=1,7,24,32,0,0,0,
G64_183=C16 x C2 x C2,N0=1,7,8,16,32,0,0,
G64_192=C4 x C4 x C2 x C2,N0=1,15,48,0,0,0,0,
G64_246=C8 x C2 x C2 x C2,N0=1,15,16,32,0,0,0,
G64_260=C4 x C2 x C2 x C2 x C2,N0=1,31,32,0,0,0,0,
G64_267=C2 x C2 x C2 x C2 x C2 x C2,N0=1,63,0,0,0,0,0,

32阶环的编号空间：
特征为16、加法群为C_16×C_2的32阶环分配编号空间7~36（已找到的上限值）

特征为8、加法群为C_8×C_4的32阶环分配编号空间1000~1115（已找到的上限值）

特征为8、加法群为C_8×C_2×C_2的32阶环分配编号空间2000~2172（已找到的上限值）

特征为4、加法群为C_4×C_4×C_2的32阶环分配编号空间3000~3999、7000~7181（已找到的上限值）
特征为4、加法群为C_4×C_2×C_2×C_2的32阶环分配编号空间4000~4999、6000~6297（已找到的上限值）
特征为2、加法群为C_2×C_2×C_2×C_2×C_2的32阶环分配编号空间5000~5607（已找到的上限值）
8000~18590的编号空间暂时不用。

64阶环的编号空间：
特征为32、加法群为C_32×C_2的64阶环分配编号空间8~10000（近似上限值）

特征为16、加法群为C_16×C_4的64阶环分配编号空间10000~20000（近似上、下限值）

特征为16、加法群为C_16×C_2×C_2的64阶环分配编号空间20000~30000（近似上、下限值）

特征为8、加法群为C_8×C_8的64阶环分配编号空间30000~40000（近似下限值）
特征为8、加法群为C_8×C_4×C_2的64阶环分配编号空间40000~50000（近似下限值）
特征为8、加法群为C_8×C_2×C_2×C_2的64阶环分配编号空间50000~60000（近似下限值）
特征为4、加法群为C_4×C_4×C_4的64阶环分配编号空间60000~70000（近似下限值）
特征为4、加法群为C_4×C_4×C_2×C_2的64阶环分配编号空间70000~80000（近似下限值）
特征为4、加法群为C_4×C_2×C_2×C_2×C_2的64阶环分配编号空间80000~90000（近似下限值）
特征为2、加法群为C_2×C_2×C_2×C_2×C_2×C_2的64阶环分配编号空间90000~100000（近似下限值）
100000~829826的编号空间暂时不用。
128阶环的编号空间：
gap> n:=128;;m:=NumberSmallGroups(n);;for i in [1..m] do G:=SmallGroup(n,i);;if IsAbelian(G) then L:=List(Elements(G),Order);M:=DivisorsInt(n);Print("G",n,"_",i,"=",StructureDescription(G),",N0=");for k in M do Print(Size(Positions(L,k)),",");od;Print("\\n");fi;od;
1~8:G128_1=C128,N0=1,1,2,4,8,16,32,64,
9~10000:G128_42=C16 x C8,N0=1,3,12,48,64,0,0,0,
10000~20000:G128_128=C32 x C4,N0=1,3,12,16,32,64,0,0,
20000~30000:G128_159=C64 x C2,N0=1,3,4,8,16,32,64,0,
30000~40000:G128_179=C8 x C8 x C2,N0=1,7,24,96,0,0,0,0,
40000~50000:G128_456=C8 x C4 x C4,N0=1,7,56,64,0,0,0,0,
50000~60000:G128_837=C16 x C4 x C2,N0=1,7,24,32,64,0,0,0,
60000~70000:G128_988=C32 x C2 x C2,N0=1,7,8,16,32,64,0,0,
70000~80000:G128_997=C4 x C4 x C4 x C2,N0=1,15,112,0,0,0,0,0,
80000~90000:G128_1601=C8 x C4 x C2 x C2,N0=1,15,48,64,0,0,0,0,
90000~100000:G128_2136=C16 x C2 x C2 x C2,N0=1,15,16,32,64,0,0,0,
100000~110000:G128_2150=C4 x C4 x C2 x C2 x C2,N0=1,31,96,0,0,0,0,0,
110000~120000:G128_2301=C8 x C2 x C2 x C2 x C2,N0=1,31,32,64,0,0,0,0,
120000~130000:G128_2319=C4 x C2 x C2 x C2 x C2 x C2,N0=1,63,64,0,0,0,0,0,
130000~140000:G128_2328=C2 x C2 x C2 x C2 x C2 x C2 x C2,N0=1,127,0,0,0,0,0,0,
140000以上的编号空间暂时不用。
256阶环的编号空间：
gap> n:=256;;m:=NumberSmallGroups(n);;for i in [1..m] do G:=SmallGroup(n,i);;if IsAbelian(G) then L:=List(Elements(G),Order);M:=DivisorsInt(n);Print("G",n,"_",i,"=",StructureDescription(G),",N0=");for k in M do Print(Size(Positions(L,k)),",");od;Print("\\n");fi;od;
1~9:G256_1=C256,N0=1,1,2,4,8,16,32,64,128,
10~10000:G256_39=C16 x C16,N0=1,3,12,48,192,0,0,0,0,
10000~20000:G256_316=C32 x C8,N0=1,3,12,48,64,128,0,0,0,
20000~30000:G256_497=C64 x C4,N0=1,3,12,16,32,64,128,0,0,
30000~40000:G256_537=C128 x C2,N0=1,3,4,8,16,32,64,128,0,
40000~50000:G256_826=C8 x C8 x C4,N0=1,7,56,192,0,0,0,0,0,
50000~60000:G256_4384=C16 x C8 x C2,N0=1,7,24,96,128,0,0,0,0,
60000~70000:G256_5525=C16 x C4 x C4,N0=1,7,56,64,128,0,0,0,0,
70000~80000:G256_6534=C32 x C4 x C2,N0=1,7,24,32,64,128,0,0,0,
80000~90000:G256_6723=C64 x C2 x C2,N0=1,7,8,16,32,64,128,0,0,
90000~100000:G256_6732=C4 x C4 x C4 x C4,N0=1,15,240,0,0,0,0,0,0,
100000~110000:G256_10298=C8 x C8 x C2 x C2,N0=1,15,48,192,0,0,0,0,0,
110000~120000:G256_13313=C8 x C4 x C4 x C2,N0=1,15,112,128,0,0,0,0,0,
120000~130000:G256_26308=C16 x C4 x C2 x C2,N0=1,15,48,64,128,0,0,0,0,
130000~140000:G256_26959=C32 x C2 x C2 x C2,N0=1,15,16,32,64,128,0,0,0,
140000~150000:G256_26973=C4 x C4 x C4 x C2 x C2,N0=1,31,224,0,0,0,0,0,0,
150000~160000:G256_53038=C8 x C4 x C2 x C2 x C2,N0=1,31,96,128,0,0,0,0,0,
160000~170000:G256_55608=C16 x C2 x C2 x C2 x C2,N0=1,31,32,64,128,0,0,0,0,
170000~180000:G256_55626=C4 x C4 x C2 x C2 x C2 x C2,N0=1,63,192,0,0,0,0,0,0,
180000~190000:G256_56059=C8 x C2 x C2 x C2 x C2 x C2,N0=1,63,64,128,0,0,0,0,0,
190000~200000:G256_56082=C4 x C2 x C2 x C2 x C2 x C2 x C2,N0=1,127,128,0,0,0,0,0,0,
200000~210000:G256_56092=C2 x C2 x C2 x C2 x C2 x C2 x C2 x C2,N0=1,255,0,0,0,0,0,0,0,
210000以上的编号空间暂时不用。