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电算游戏(六)“901”型的等式队列

(2010-10-04 07:30:53)
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分类: 科普

(六)“901”型的等式队列

6·1 寻访“901”型数

前面已经见过以数字“9”为首位,以“1”收尾的自然数,如909091,9901,等等,我们用“901” 来称呼这一类型的数,如九联环解套游戏,也可以俗称九联环数。在纯元数F(m)=111…11(m个1)的质因数分解中,“901”型的质数出现不少,总是跟回文数“结缘”:

F(10)=122221×9091

F(14)=12222221×909091

F(20)=1222210000122221×9091

F(24)=112233332211×9091×99990001

F(28)=1222222100000012222221×909091

F(36)=333000333333000333×999999000001

F(38)=12222222222222222221×909090909090909091

F(39)=123333333333321×900900900900990990990991

F(48)=11223333333333332211×9901×99990001×9999999900000001

F(62)=12222222222222222222222222222221×909090909090909090909090909091

F(72)=112233444444332210887766667789112233444444332211×9901×99990001×999999000001

F(76)=1222222222222222222100000000000000000012222222222222222221×909090909090909091

F(78)=123333333333321000000000000000000000000123333333333321×900900900900990990990991

F(93)=123333333333333333333333333333321

     ×900900900900900900900900900900990990990990990990990990990991

F(96)=11223333333333332211000000000000000000000000000011223333333333332211

     ×9901×99990001×9999999900000001

F106)=122222222222222222222222222222222222222222222222222221

     ×9090909090909090909090909090909090909090909090909091

F(111)=123333333333333333333333333333333333321

      ×900900900900900900900900900900900900990990990990990990990990990990990991

F(115)=123455555555555555555554321

     ×90000900009000090000900909009090090900909009099090990909909099090

                                                99099990999909999099991

F(117)=123333333333321000000000000000000000000123333333333321000000000000000000000000

                                        123333333333321×900900900900990990990991

F(119)=12345677777777777654321

×900000090000009009000900900090090099009009900900990990099099009909909990990999099099999909999991

F(123)=1233333333333333333333333333333333333333321

    ×90090090090090090090090090090090090090090990990990990990990990990990990990990991

每一个等式都可以建构一个等式队列!

上列各个纯元数分解的等式中,出现的质数有:9091,909091,909090909090909091,90909090909090909090909090909(14个90,是F(62)的质因数),

9090909090909090909090909090909090909090909090909091(25个90,是F(106)的质因数);

在F(48)、F(72) 与F(96)分解式中各出现三个质因数:99990001,999999000001与9999999900000001;

F(39) 、F(78) 与F(117) 分解式中出现质因数900900900900990990990991

F(93)与F(117)分解式中分别出现质因数:

900900900900900900900900900900990990990990990990990990990991与

900900900900900900900900900900900900990990990990990990990990990990990991;

    F(115)与F(119)分解式中分别出现因数:

9000090000900009000090090900909009090090900909909099090990909909099099990999909999099991与

900000090000009009000900900090090099009009900900990990099099009909909990990999099099999909999991

上述分解等式有的是创建等式队列和数字大厦的很好素材如:

1233321×90090991111111111111111

12344321×90009900999111111111111111111111

123333321×900900990991111111111111111111111

1234444321×9000900990099099911111111111111111111111111111

1222222222222221×90909090909091111111111111111111111111111111

12345554321×90000909009090990909999111111111111111111111111111111111111

123456654321×900000990000999000999900999991111111111111111111111111111111111111111111

12345677654321×900000099000009990000999900099999009999991

11111111111111111111111111111111111111111111111111111111

前面见过的“缺8数”,衍生出的回文数有的是质数,如: 123456797654321,1235321,12421,131是质数。而有的不是质数,如:1234568654321=277169×4454209,12345754321=23×223×2407049,123464321=389×433×733。这些回文数经过特殊的九联环变幻,仍然还是回文数,有如下等式队列:

1235321×909111230303211

123464321×90911122414142211

12345754321×9091112235252532211

1234568654321×909111223463636432211

123456797654321×90911122345747475432211

123456797654321×909091112233463636364332211

123456797654321×9090919111223345252525254332211

123456797654321×90909090911122334524141414254332211

123456797654321×909090909091112233452413030314254332211

123456797654321×9111234568586543211

123456797654321×99011222345753575432221

123456797654321×999001123333464313464333321

123456797654321×9900990112223445313531354432221

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