《数论探秘》电子版

ysr

代码如下：
Private Sub Command1_Click()
Dim a, b
a1 = Val(Text1)
a = 2
Do While a <= a1
b = 1
Do While b <= a1
x = a + b
y = a - b
z = a
p = 2 * b
q = a ^ 2 + b ^ 2
X1 = x ^ (1 / 2): Y1 = Abs(y) ^ (1 / 2): z1 = z ^ (1 / 2): p1 = p ^ (1 / 2): q1 = q ^ (1 / 2)
If InStr(Y1, ".") = 0 And InStr(z1, ".") = 0 And InStr(X1, ".") = 0 And InStr(X1, ".") = 0 Then
s = s + 1
s1 = s1 & "a=" & a & " b=" & b & " x=" & x & " y=" & y & " z=" & z & " p=" & p & " q=" & q & vbCrLf
Else
s = s
End If

b = b + 1
Loop


a = a + 1
Loop

If s > 0 Then
Text2 = "a1=" & a1 & " s=" & s & vbCrLf & s1
Else
Text2 = "wu   jie"
End If

End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""

End Sub

ysr

哥德巴赫猜想并不难，有多种初等证明方法可以证明，哥德巴赫猜想是远远成立的。
1，由差定理（更容易证明）证明和定理（就是哥德巴赫猜想）成立。
2，设偶数2A的方根为M则其方根M内的素数的个数的下限是m=M/lnM，则偶数2A的哥德巴赫猜想解的个数的绝对下限就是m-1，这是对无穷大的偶数都成立的，随着偶数的增大实际解的个数远远大于m-1 ， 所以，哥德巴赫猜想远远成立。
3，据构成哥德巴赫猜想解的素数与偶数的方根的大小，把解分为两类：小根拆和大根拆，大于4的偶数，仅仅有73个偶数只有大根拆而不含有小根拆，其他的都是既有小根拆也有大根拆，而4=2+2.

所以，哥德巴赫猜想远远成立，容易证明，仅仅初等数学就可以证明，中学以上的学历都i可以完全解决。

至今不能解决的原因仅仅有两个：一是数学家喜欢本末倒置从解析数论下手解决问题，二是中国数学界到处是汉奸洋奴才等破环了中国数学界的学术氛围！！

ysr

1267650600228229401496703254299内有s=2个：
1267650600228229401496703251307   1267650600228229401496703251309  n1= 158456325028528675187087906413
1267650600228229401496703253347   1267650600228229401496703253349  n1= 158456325028528675187087906668

这是连续1000项内找到的两对孪生素数

ysr

我的《数论探秘》在淘宝上有销售，价格也不贵因为是首发，点开淘宝，搜索“筑书文化”店铺，点击进入店铺，就可以看到我的《数论探秘》了。欢迎浏览和购买，欢迎结缘！

希望普及知识 ，弘扬科学精神！
我的书得到普及的话，由于用到的知识点儿不高，有关知识将有可能很快会进入中学课本。

我有著作权登记证，受中国知识产权保护中心保护呢，虽然不能证明我的知识点儿高级，但是能证明我的论文不与别人的学术重复是受保护的。

ysr

前天发某院快递内包装一本新书我的《数论探秘》，请该院转交一位在数学方面有影响力的名人，软件显示9.16日上午10:21已经签收，可是我到现在也没收到回复  ，谁知道是不是没有给转交？据网友介绍该院有俩人处理这样的事，该俩人每天会往收费站卖大量废纸，可能我的书已经送往收费站了。
中国人发表一个成果何其艰难？似这样下去如何才能发展科技？

ysr

一种高效的计算素数的方法：埃拉托斯特尼筛法（Sieve of Eratosthenes）

asdfg1258963

于 2025-04-01 11:39:47 发布

阅读量640
收藏 10

点赞数 21
CC 4.0 BY-SA版权
分类专栏： 数据结构知识 文章标签： 算法 python

数据结构知识
专栏收录该内容
7 篇文章
订阅专栏
素数筛
一、基本思想
通过标记非素数（即合数），从而得到所有素数。

二、算法步骤
初始化：

创建一个布尔型列表 sieve，表示从 0 到 n 每个数是否是素数。初始时，我们假设所有数都是素数，因此将列表中所有的元素都设置为 True，但 sieve[0] 和 sieve[1] 设为 False，因为 0 和 1 不是素数。
筛选过程：

从 2 开始，遍历到 sqrt(n)，&#11088;因为如果一个合数 a 可以被分解成两个因数 b 和 c，则必定存在 b 或 c 小于等于 sqrt(a)。因此，我们只需要检查小于等于 sqrt(n) 的数，来筛选出所有的合数。
对于每一个被标记为素数的数 i，我们将 i 的倍数（即从 i*i 开始的所有 i 的倍数）标记为合数，即将 sieve[j] 设置为 False。
收集素数：

遍历 sieve 列表，将其中值为 True 的索引（即素数）收集到一个新列表 primes 中。
&#11088;为什么从 i * i 开始标记？
标记合数的过程： 在算法中，i 是当前的素数。如果 i 是素数，那么我们从 i * i 开始标记，因为：

小于 i 的倍数（即 2i, 3i, …）已经在之前的素数处理中标记过了。
从 i * i 开始标记，可以确保我们不会重复标记已经处理过的合数，而且可以避免不必要的操作。
避免重复计算：

如果 i 是素数，那么 i 的倍数 2i, 3i, 4i, ... 已经在之前的步骤中被标记过了（因为这些倍数的因数已经小于 i，且我们在处理更小的素数时已经将它们标记为合数）。
因此，开始从 i * i 标记是为了跳过那些已经被处理过的倍数，避免重复操作，从而提高效率。
三、例子：
假设我们计算素数 n = 30，我们要找出小于等于 30 的所有素数。

从 i = 2 开始，2 是素数，标记 2 的倍数：4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30。
然后 i = 3，3 也是素数，标记 3 的倍数：9, 12, 15, 18, 21, 24, 27, 30。
接下来，i = 4，sieve[4] 已经是 False（在处理 2 时就已经标记过了），跳过。
然后 i = 5，5 是素数，标记 5 的倍数：25, 30。
注意到 5 * 5 = 25，这是第一次标记 5 的倍数，而更小的倍数（如 5 * 2 = 10）已经在前面标记过了。
通过这种方式，我们避免了重复的计算，提升了效率。

四、核心代码
def sieve_of_eratosthenes(n):
    # 初始化布尔数组，假设所有数都是素数
    sieve = [True] * (n + 1)  # 创建大小为 n+1 的布尔数组
    sieve[0] = sieve[1] = False  # 0 和 1 不是素数

    # 从 2 开始筛选
    for i in range(2, int(n ** 0.5) + 1):  # 只需到 sqrt(n)&#11088;转换为int类型
        if sieve[i]:  # 如果 i 是素数
            for j in range(i * i, n + 1, i):  # 将 i 的倍数标记为合数
                sieve[j] = False
   
    # 返回所有值为 True 的索引，这些索引对应的数字是素数
    primes = [i for i in range(2, n + 1) if sieve[i]]
    return primes

ysr

>>> def sieve_of_eratosthenes(n):
    # 初始化布尔数组，假设所有数都是素数
         sieve = [True] * (n + 1)  # 创建大小为 n+1 的布尔数组
         sieve[0] = sieve[1] = False  # 0 和 1 不是素数

    # 从 2 开始筛选
         for i in range(2, int(n ** 0.5) + 1):  # 只需到 sqrt(n)&#11088;转换为int类型
            if sieve[i]:  # 如果 i 是素数
                for j in range(i * i, n + 1, i):  # 将 i 的倍数标记为合数
                    sieve[j] = False
   
    # 返回所有值为 True 的索引，这些索引对应的数字是素数
         primes = [i for i in range(2, n + 1) if sieve[i]]
         return primes

        
>>> print(sieve_of_eratosthenes(30))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
>>> print(sieve_of_eratosthenes(100))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
>>> print(sieve_of_eratosthenes(1000))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]
>>>

ysr

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 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ysr

10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149, 11159, 11161, 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287, 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399, 11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497, 11503, 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, 12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591, 13597, 13613, 13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687, 13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879, 13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13997, 13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, 14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243, 14249, 14251, 14281, 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369, 14387, 14389, 14401, 14407, 14411, 14419, 14423, 14431, 14437, 14447, 14449, 14461, 14479, 14489, 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557, 14561, 14563, 14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, 14657, 14669, 14683, 14699, 14713, 14717, 14723, 14731, 14737, 14741, 14747, 14753, 14759, 14767, 14771, 14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843, 14851, 14867, 14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947, 14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077, 15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, 15161, 15173, 15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269, 15271, 15277, 15287, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359, 15361, 15373, 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581, 15583, 15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881, 15887, 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971, 15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187, 16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, 16273, 16301, 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417, 16421, 16427, 16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, 16547, 16553, 16561, 16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651, 16657, 16661, 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759, 16763, 16787, 16811, 16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903, 16921, 16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987, 16993, 17011, 17021, 17027, 17029, 17033, 17041, 17047, 17053, 17077, 17093, 17099, 17107, 17117, 17123, 17137, 17159, 17167, 17183, 17189, 17191, 17203, 17207, 17209, 17231, 17239, 17257, 17291, 17293, 17299, 17317, 17321, 17327, 17333, 17341, 17351, 17359, 17377, 17383, 17387, 17389, 17393, 17401, 17417, 17419, 17431, 17443, 17449, 17467, 17471, 17477, 17483, 17489, 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, 17597, 17599, 17609, 17623, 17627, 17657, 17659, 17669, 17681, 17683, 17707, 17713, 17729, 17737, 17747, 17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839, 17851, 17863, 17881, 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957, 17959, 17971, 17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049, 18059, 18061, 18077, 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143, 18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233, 18251, 18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313, 18329, 18341, 18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433, 18439, 18443, 18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521, 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637, 18661, 18671, 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749, 18757, 18773, 18787, 18793, 18797, 18803, 18839, 18859, 18869, 18899, 18911, 18913, 18917, 18919, 18947, 18959, 18973, 18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073, 19079, 19081, 19087, 19121, 19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211, 19213, 19219, 19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319, 19333, 19373, 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427, 19429, 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477, 19483, 19489, 19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571, 19577, 19583, 19597, 19603, 19609, 19661, 19681, 19687, 19697, 19699, 19709, 19717, 19727, 19739, 19751, 19753, 19759, 19763, 19777, 19793, 19801, 19813, 19819, 19841, 19843, 19853, 19861, 19867, 19889, 19891, 19913, 19919, 19927, 19937, 19949, 19961, 19963, 19973, 19979, 19991, 19993, 19997, 20011, 20021, 20023, 20029, 20047, 20051, 20063, 20071, 20089, 20101, 20107, 20113, 20117, 20123, 20129, 20143, 20147, 20149, 20161, 20173, 20177, 20183, 20201, 20219, 20231, 20233, 20249, 20261, 20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357, 20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, 20477, 20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563, 20593, 20599, 20611, 20627, 20639, 20641, 20663, 20681, 20693, 20707, 20717, 20719, 20731, 20743, 20747, 20749, 20753, 20759, 20771, 20773, 20789, 20807, 20809, 20849, 20857, 20873, 20879, 20887, 20897, 20899, 20903, 20921, 20929, 20939, 20947, 20959, 20963, 20981, 20983, 21001, 21011, 21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067, 21089, 21101, 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179, 21187, 21191, 21193, 21211, 21221, 21227, 21247, 21269, 21277, 21283, 21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379, 21383, 21391, 21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, 21491, 21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, 21559, 21563, 21569, 21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647, 21649, 21661, 21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757, 21767, 21773, 21787, 21799, 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871, 21881, 21893, 21911, 21929, 21937, 21943, 21961, 21977, 21991, 21997, 22003, 22013, 22027, 22031, 22037, 22039, 22051, 22063, 22067, 22073, 22079, 22091, 22093, 22109, 22111, 22123, 22129, 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229, 22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307, 22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433, 22441, 22447, 22453, 22469, 22481, 22483, 22501, 22511, 22531, 22541, 22543, 22549, 22567, 22571, 22573, 22613, 22619, 22621, 22637, 22639, 22643, 22651, 22669, 22679, 22691, 22697, 22699, 22709, 22717, 22721, 22727, 22739, 22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817, 22853, 22859, 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963, 22973, 22993, 23003, 23011, 23017, 23021, 23027, 23029, 23039, 23041, 23053, 23057, 23059, 23063, 23071, 23081, 23087, 23099, 23117, 23131, 23143, 23159, 23167, 23173, 23189, 23197, 23201, 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297, 23311, 23321, 23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447, 23459, 23473, 23497, 23509, 23531, 23537, 23539, 23549, 23557, 23561, 23563, 23567, 23581, 23593, 23599, 23603, 23609, 23623, 23627, 23629, 23633, 23663, 23669, 23671, 23677, 23687, 23689, 23719, 23741, 23743, 23747, 23753, 23761, 23767, 23773, 23789, 23801, 23813, 23819, 23827, 23831, 23833, 23857, 23869, 23873, 23879, 23887, 23893, 23899, 23909, 23911, 23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007, 24019, 24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103, 24107, 24109, 24113, 24121, 24133, 24137, 24151, 24169, 24179, 24181, 24197, 24203, 24223, 24229, 24239, 24247, 24251, 24281, 24317, 24329, 24337, 24359, 24371, 24373, 24379, 24391, 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 24499, 24509, 24517, 24527, 24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631, 24659, 24671, 24677, 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767, 24781, 24793, 24799, 24809, 24821, 24841, 24847, 24851, 24859, 24877, 24889, 24907, 24917, 24919, 24923, 24943, 24953, 24967, 24971, 24977, 24979, 24989, 25013, 25031, 25033, 25037, 25057, 25073, 25087, 25097, 25111, 25117, 25121, 25127, 25147, 25153, 25163, 25169, 25171, 25183, 25189, 25219, 25229, 25237, 25243, 25247, 25253, 25261, 25301, 25303, 25307, 25309, 25321, 25339, 25343, 25349, 25357, 25367, 25373, 25391, 25409, 25411, 25423, 25439, 25447, 25453, 25457, 25463, 25469, 25471, 25523, 25537, 25541, 25561, 25577, 25579, 25583, 25589, 25601, 25603, 25609, 25621, 25633, 25639, 25643, 25657, 25667, 25673, 25679, 25693, 25703, 25717, 25733, 25741, 25747, 25759, 25763, 25771, 25793, 25799, 25801, 25819, 25841, 25847, 25849, 25867, 25873, 25889, 25903, 25913, 25919, 25931, 25933, 25939, 25943, 25951, 25969, 25981, 25997, 25999, 26003, 26017, 26021, 26029, 26041, 26053, 26083, 26099, 26107, 26111, 26113, 26119, 26141, 26153, 26161, 26171, 26177, 26183, 26189, 26203, 26209, 26227, 26237, 26249, 26251, 26261, 26263, 26267, 26293, 26297, 26309, 26317, 26321, 26339, 26347, 26357, 26371, 26387, 26393, 26399, 26407, 26417, 26423, 26431, 26437, 26449, 26459, 26479, 26489, 26497, 26501, 26513, 26539, 26557, 26561, 26573, 26591, 26597, 26627, 26633, 26641, 26647, 26669, 26681, 26683, 26687, 26693, 26699, 26701, 26711, 26713, 26717, 26723, 26729, 26731, 26737, 26759, 26777, 26783, 26801, 26813, 26821, 26833, 26839, 26849, 26861, 26863, 26879, 26881, 26891, 26893, 26903, 26921, 26927, 26947, 26951, 26953, 26959, 26981, 26987, 26993, 27011, 27017, 27031, 27043, 27059, 27061, 27067, 27073, 27077, 27091, 27103, 27107, 27109, 27127, 27143, 27179, 27191, 27197, 27211, 27239, 27241, 27253, 27259, 27271, 27277, 27281, 27283, 27299, 27329, 27337, 27361, 27367, 27397, 27407, 27409, 27427, 27431, 27437, 27449, 27457, 27479, 27481, 27487, 27509, 27527, 27529, 27539, 27541, 27551, 27581, 27583, 27611, 27617, 27631, 27647, 27653, 27673, 27689, 27691, 27697, 27701, 27733, 27737, 27739, 27743, 27749, 27751, 27763, 27767, 27773, 27779, 27791, 27793, 27799, 27803, 27809, 27817, 27823, 27827, 27847, 27851, 27883, 27893, 27901, 27917, 27919, 27941, 27943, 27947, 27953, 27961, 27967, 27983, 27997, 28001, 28019, 28027, 28031, 28051, 28057, 28069, 28081, 28087, 28097, 28099, 28109, 28111, 28123, 28151, 28163, 28181, 28183, 28201, 28211, 28219, 28229, 28277, 28279, 28283, 28289, 28297, 28307, 28309, 28319, 28349, 28351, 28387, 28393, 28403, 28409, 28411, 28429, 28433, 28439, 28447, 28463, 28477, 28493, 28499, 28513, 28517, 28537, 28541, 28547, 28549, 28559, 28571, 28573, 28579, 28591, 28597, 28603, 28607, 28619, 28621, 28627, 28631, 28643, 28649, 28657, 28661, 28663, 28669, 28687, 28697, 28703, 28711, 28723, 28729, 28751, 28753, 28759, 28771, 28789, 28793, 28807, 28813, 28817, 28837, 28843, 28859, 28867, 28871, 28879, 28901, 28909, 28921, 28927, 28933, 28949, 28961, 28979, 29009, 29017, 29021, 29023, 29027, 29033, 29059, 29063, 29077, 29101, 29123, 29129, 29131, 29137, 29147, 29153, 29167, 29173, 29179, 29191, 29201, 29207, 29209, 29221, 29231, 29243, 29251, 29269, 29287, 29297, 29303, 29311, 29327, 29333, 29339, 29347, 29363, 29383, 29387, 29389, 29399, 29401, 29411, 29423, 29429, 29437, 29443, 29453, 29473, 29483, 29501, 29527, 29531, 29537, 29567, 29569, 29573, 29581, 29587, 29599, 29611, 29629, 29633, 29641, 29663, 29669, 29671, 29683, 29717, 29723, 29741, 29753, 29759, 29761, 29789, 29803, 29819, 29833, 29837, 29851, 29863, 29867, 29873, 29879, 29881, 29917, 29921, 29927, 29947, 29959, 29983, 29989, 30011, 30013, 30029, 30047, 30059, 30071, 30089, 30091, 30097, 30103, 30109, 30113, 30119, 30133, 30137, 30139, 30161, 30169, 30181, 30187, 30197, 30203, 30211, 30223, 30241, 30253, 30259, 30269, 30271, 30293, 30307, 30313, 30319, 30323,

ysr

30341, 30347, 30367, 30389, 30391, 30403, 30427, 30431, 30449, 30467, 30469, 30491, 30493, 30497, 30509, 30517, 30529, 30539, 30553, 30557, 30559, 30577, 30593, 30631, 30637, 30643, 30649, 30661, 30671, 30677, 30689, 30697, 30703, 30707, 30713, 30727, 30757, 30763, 30773, 30781, 30803, 30809, 30817, 30829, 30839, 30841, 30851, 30853, 30859, 30869, 30871, 30881, 30893, 30911, 30931, 30937, 30941, 30949, 30971, 30977, 30983, 31013, 31019, 31033, 31039, 31051, 31063, 31069, 31079, 31081, 31091, 31121, 31123, 31139, 31147, 31151, 31153, 31159, 31177, 31181, 31183, 31189, 31193, 31219, 31223, 31231, 31237, 31247, 31249, 31253, 31259, 31267, 31271, 31277, 31307, 31319, 31321, 31327, 31333, 31337, 31357, 31379, 31387, 31391, 31393, 31397, 31469, 31477, 31481, 31489, 31511, 31513, 31517, 31531, 31541, 31543, 31547, 31567, 31573, 31583, 31601, 31607, 31627, 31643, 31649, 31657, 31663, 31667, 31687, 31699, 31721, 31723, 31727, 31729, 31741, 31751, 31769, 31771, 31793, 31799, 31817, 31847, 31849, 31859, 31873, 31883, 31891, 31907, 31957, 31963, 31973, 31981, 31991, 32003, 32009, 32027, 32029, 32051, 32057, 32059, 32063, 32069, 32077, 32083, 32089, 32099, 32117, 32119, 32141, 32143, 32159, 32173, 32183, 32189, 32191, 32203, 32213, 32233, 32237, 32251, 32257, 32261, 32297, 32299, 32303, 32309, 32321, 32323, 32327, 32341, 32353, 32359, 32363, 32369, 32371, 32377, 32381, 32401, 32411, 32413, 32423, 32429, 32441, 32443, 32467, 32479, 32491, 32497, 32503, 32507, 32531, 32533, 32537, 32561, 32563, 32569, 32573, 32579, 32587, 32603, 32609, 32611, 32621, 32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717, 32719, 32749, 32771, 32779, 32783, 32789, 32797, 32801, 32803, 32831, 32833, 32839, 32843, 32869, 32887, 32909, 32911, 32917, 32933, 32939, 32941, 32957, 32969, 32971, 32983, 32987, 32993, 32999, 33013, 33023, 33029, 33037, 33049, 33053, 33071, 33073, 33083, 33091, 33107, 33113, 33119, 33149, 33151, 33161, 33179, 33181, 33191, 33199, 33203, 33211, 33223, 33247, 33287, 33289, 33301, 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34729, 34739, 34747, 34757, 34759, 34763, 34781, 34807, 34819, 34841, 34843, 34847, 34849, 34871, 34877, 34883, 34897, 34913, 34919, 34939, 34949, 34961, 34963, 34981, 35023, 35027, 35051, 35053, 35059, 35069, 35081, 35083, 35089, 35099, 35107, 35111, 35117, 35129, 35141, 35149, 35153, 35159, 35171, 35201, 35221, 35227, 35251, 35257, 35267, 35279, 35281, 35291, 35311, 35317, 35323, 35327, 35339, 35353, 35363, 35381, 35393, 35401, 35407, 35419, 35423, 35437, 35447, 35449, 35461, 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537, 35543, 35569, 35573, 35591, 35593, 35597, 35603, 35617, 35671, 35677, 35729, 35731, 35747, 35753, 35759, 35771, 35797, 35801, 35803, 35809, 35831, 35837, 35839, 35851, 35863, 35869, 35879, 35897, 35899, 35911, 35923, 35933, 35951, 35963, 35969, 35977, 35983, 35993, 35999, 36007, 36011, 36013, 36017, 36037, 36061, 36067, 36073, 36083, 36097, 36107, 36109, 36131, 36137, 36151, 36161, 36187, 36191, 36209, 36217, 36229, 36241, 36251, 36263, 36269, 36277, 36293, 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