有多少1999类型的素数？
给孩子妈妈转账，不想转2000，就转了1999，
结果我仔细研究了一下，
发现1999是质数，
再研究发展2999,4999,8999都是素数。
19,29,59,79,89都是素数，
手边没电脑，这类素数还有哪些？ 看上去挺多的
19
29
59
79
89
199
499
599
1999
2999
4999
8999
49999
59999
79999
199999
599999
799999
2999999
4999999
19999999
29999999
59999999
89999999
799999999
59999999999
79999999999
59999999999999
499999999999999
29999999999999999999
89999999999999999999
59999999999999999999999
599999999999999999999999
79999999999999999999999999
199999999999999999999999999
1999999999999999999999999999
2999999999999999999999999999
59999999999999999999999999999
899999999999999999999999999999
59999999999999999999999999999999999
89999999999999999999999999999999999999
59999999999999999999999999999999999999999
29999999999999999999999999999999999999999999
79999999999999999999999999999999999999999999999999
199999999999999999999999999999999999999999999999999999
4999999999999999999999999999999999999999999999999999999
29999999999999999999999999999999999999999999999999999999
59999999999999999999999999999999999999999999999999999999999999
59999999999999999999999999999999999999999999999999999999999999999999999999
79999999999999999999999999999999999999999999999999999999999999999999999999999 mathe 发表于 2025-3-29 17:35
看上去挺多的
19
29
你的代码怎么弄的？
亮出你的代码！ nyy 发表于 2025-3-29 22:26
你的代码怎么弄的？
亮出你的代码！
ieos有这串数，只是人家也吝啬，没有任何软件的代码。
A141311
Primes consisting of a digit and a nonempty string of 9's (i.e., primes of the form k*10^m - 1, where k is any digit).
由一个数字和一个9的非空字符串组成的质数（即，k*10^m - 1的形式的质数，其中k是任何数字）。
+30 30
5
19, 29, 59, 79, 89, 199, 499, 599, 1999, 2999, 4999, 8999, 49999, 59999, 79999, 199999, 599999, 799999, 2999999, 4999999, 19999999, 29999999, 59999999, 89999999, 799999999, 59999999999, 79999999999, 59999999999999, 499999999999999, 29999999999999999999
特别：k*10^m - 1，K永远不可能是1 4或7，因为如果是，K *10^m - 1就能被3整除。
n的表，a(n)对于n = 1..71（下一项是8*10^1219 - 1），数列中研究到第71项。 本帖最后由 数论爱好者 于 2025-3-30 09:23 编辑
数论爱好者 发表于 2025-3-30 05:51
ieos有这串数，只是人家也吝啬，没有任何软件的代码。
A141311
Primes consisting of a digit and a none ...
AI写了一段maple代码，已经运行成功。
restart;
k_values := :# 抑制输出
output_file := "primes_results.txt":
fileID := fopen(output_file, WRITE):
try
fprintf(fileID, "Primes of the form k*10^m -1 (k≠1,4,7; m=1..2000):\n\n");
for k in k_values do
printf("----- Scanning k = %d -----\n", k);
start_time := time();
for m from 1 to 2000 do
n := k * 10^m - 1;
if isprime(n) then
# 按指定格式写入文件
fprintf(fileID, "k=%d, m=%d, p=%a\n", k, m, n);
# 控制台同步输出
printf("Found: k=%d, m=%d, p=%a\n", k, m, n);
end if;
end do;
printf("k=%d completed in %.2f seconds\n\n", k, time()-start_time);
end do;
fprintf(fileID, "\nEnd of results");
finally
fclose(fileID);
end try:
# 打印最终结果到屏幕
printf("\n=== 最终结果（文件内容预览） ===\n");
readlines(output_file); $k={2, 3, 5, 6, 8, 9},p=k*10^m-1$
n=0;Do;If,n++;Print[{n,{k,m}}]],{m,10000},{r,6}]
{1,{2,1}}
{2,{3,1}}
{3,{6,1}}
{4,{8,1}}
{5,{9,1}}
{6,{2,2}}
{7,{5,2}}
{8,{6,2}}
{9,{2,3}}
{10,{3,3}}
{11,{5,3}}
{12,{9,3}}
{13,{5,4}}
{14,{6,4}}
{15,{8,4}}
{16,{2,5}}
{17,{6,5}}
{18,{8,5}}
{19,{3,6}}
{20,{5,6}}
{21,{2,7}}
{22,{3,7}}
{23,{6,7}}
{24,{9,7}}
{25,{8,8}}
{26,{6,10}}
{27,{8,10}}
{28,{6,13}}
{29,{5,14}}
{30,{3,19}}
{31,{9,19}}
{32,{6,22}}
{33,{6,23}}
{34,{8,25}}
{35,{2,26}}
{36,{2,27}}
{37,{3,27}}
{38,{6,28}}
{39,{9,29}}
{40,{6,34}}
{41,{9,37}}
{42,{6,40}}
{43,{3,43}}
{44,{8,49}}
{45,{2,53}}
{46,{5,54}}
{47,{3,55}}
{48,{6,61}}
{49,{6,73}}
{50,{8,76}}
{51,{9,93}}
{52,{8,128}}
{53,{2,147}}
{54,{8,175}}
{55,{3,207}}
{56,{5,210}}
{57,{2,236}}
{58,{8,238}}
{59,{2,248}}
{60,{6,361}}
{61,{2,386}}
{62,{5,390}}
{63,{2,401}}
{64,{6,490}}
{65,{2,546}}
{66,{8,550}}
{67,{5,594}}
{68,{6,613}}
{69,{2,785}}
{70,{8,796}}
{71,{9,935}}
{72,{8,1219}}
{73,{3,1311}}
{74,{2,1325}}
{75,{6,1624}}
{76,{2,1755}}
{77,{6,2000}}
{78,{8,2012}}
{79,{8,2846}}
{80,{2,2906}}
{81,{6,2994}}
{82,{2,3020}}
{83,{3,3204}}
{84,{5,3460}}
{85,{6,4301}}
{86,{6,4332}}
{87,{5,5028}}
{88,{5,5219}}
{89,{5,5332}}
{90,{2,5407}}
{91,{2,5697}}
{92,{2,5969}}
{93,{3,7050}}
{94,{2,7517}}
{95,{5,8072}}
{96,{9,8415}}
{97,{3,9439}}
{98,{9,9631}} northwolves 发表于 2025-3-30 12:21
$k={2, 3, 5, 6, 8, 9},p=k*10^m-1$
Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
(*生成可能的这种形式的数*)
aaa=Flatten@Table[(k+1)*10^n-1,{n,1,20},{k,{1,2,4,5,7,8}}]
bbb=Select&](*选出其中的数据*)
ccc=Transpose@{bbb}(*搞成列矩阵*)
Grid(*列表显示*)
\[\begin{array}{l}
19 \\
29 \\
59 \\
79 \\
89 \\
199 \\
499 \\
599 \\
1999 \\
2999 \\
4999 \\
8999 \\
49999 \\
59999 \\
79999 \\
199999 \\
599999 \\
799999 \\
2999999 \\
4999999 \\
19999999 \\
29999999 \\
59999999 \\
89999999 \\
799999999 \\
59999999999 \\
79999999999 \\
59999999999999 \\
499999999999999 \\
29999999999999999999 \\
89999999999999999999 \\
\end{array}\]
19
29
59
79
89
199
499
599
1999
2999
4999
8999
49999
59999
79999
199999
599999
799999
2999999
4999999
19999999
29999999
59999999
89999999
799999999
59999999999
79999999999
59999999999999
499999999999999
29999999999999999999
89999999999999999999
这些是前面的几个结果
1999,2999,4999,8999
这个1,2,4,8居然还是等比数列 nyy 发表于 2025-3-31 08:38
1999,2999,4999,8999
这个1,2,4,8居然还是等比数列
找出2^k*1000+999数列中是素数的情况。
Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
aaa=Table[{k,2^k*1000+999},{k,0,100}]
bbb=Select]]&](*选出第二列元素是素数的情况*)
Grid(*列表显示*)
输出结果
0 1999
1 2999
2 4999
3 8999
5 32999
9 512999
12 4096999
13 8192999
17 131072999
20 1048576999
22 4194304999
27 134217728999
28 268435456999
32 4294967296999
35 34359738368999
64 18446744073709551616999
72 4722366482869645213696999
73 9444732965739290427392999
100 1267650600228229401496703205376999
Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
(*生成可能的这种形式的数*)
aaa=Flatten[#,1]&@Table[{n,(k+1)*10^n-1},{n,1,20},{k,{1,2,4,5,7,8}}]
bbb=Select]]&](*选出其中的数据*)
Grid(*列表显示*)
求解结果
1 19
1 29
1 59
1 79
1 89
2 199
2 499
2 599
3 1999
3 2999
3 4999
3 8999
4 49999
4 59999
4 79999
5 199999
5 599999
5 799999
6 2999999
6 4999999
7 19999999
7 29999999
7 59999999
7 89999999
8 799999999
10 59999999999
10 79999999999
13 59999999999999
14 499999999999999
19 29999999999999999999
19 89999999999999999999
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