“…”型数及其等式队列

2．“9090…91”型数及其等式队列

由n（n＝1，2，3，…）个90顺次连接且以91收尾的数称作“9090…91”型数．我们以出现“90”的个数作为“9090…91”型数的标志，作为特例，我们不妨将91归入这一类型的数．“9090…91”型数不是对称数，却是对称数的再造者．我们有等式队列：

11×9091＝100001

121×9091＝1100011

12321×9091＝112010211

1234321×9091＝11221212211

123454321×9091＝1122323232211

12345654321×9091＝112234343432211

1234567654321×9091＝11223454545432211

123456787654321×9091＝1122345656565432211

12345678987654321×9091＝112234567676765432211

12345678987654321×909091＝11223345656565654332211

12345678987654321×90909091＝1122334454545454544332211

12345678987654321×9090909091＝112233445343434343544332211

12345678987654321×909090909091＝11223344534232323243544332211

12345678987654321×90909090909091＝1122334453423121213243544332211

12345678987654321×9090909090909091＝112233445342312010213243544332211

123456789987654321×909090909090909091＝112233445443322110011223344544332211

1234567899987654321×909090909090909091＝1122334454534231201021324354544332211

12345678999987654321×909090909090909091＝11223344545443322111122334454544332211

123456789999987654321×909090909090909091＝1122334454544332211001122334454544332211

1234567899999987654321×909090909090909091＝112233445454433221100001122334454544332211

从上述等式队列中任取一个等式，都可以扩展为有趣的对称数等式队列，如：

123456789987654321×9091＝1122345677777765432211

123456789987654321×909091＝112233456666666654332211

123456789987654321×90909091＝11223344555555555544332211

123456789987654321×9090909091＝1122334454444444444544332211

123456789987654321×909090909091＝112233445443333333344544332211

123456789987654321×90909090909091＝11223344544332222223344544332211

123456789987654321×9090909090909091＝1122334454433221111223344544332211

在纯元数的分解中，对出现“9090…91”型的分解式进行变换，可以看到对称数经过“9090…91”型的数的整合，又会得到纯元数等式队列：

1001×91＝111111

122221×9091＝1111111111

11222211×9901＝111111111111

12222221×909091＝11111111111111

1222222221×90909091＝111111111111111111

12222100001222211×9091＝11111111111111111111

122222222221×9090909091＝1111111111111111111111

1222222100000012222221×909091＝1111111111111111111111111111

12222100001222210000122221×9091＝111111111111111111111111111111

122222222222222221×9090909090909091＝1111111111111111111111111111111111

11222211000011222211000011222211×9901＝111111111111111111111111111111111111

12222222222222222221×909090909090909091＝11111111111111111111111111111111111111

122221000012222100001222210000122221×9091＝1111111111111111111111111111111111111111

122222210000001222222100000012222221×909091＝111111111111111111111111111111111111111111

1222222222210000000000122222222221×9090909091＝11111111111111111111111111111111111111111111

122222222222222222222221×9090909090909090909091＝1111111111111111111111111111111111111111111111

11222211000011222211000011222211000011222211×9901＝111111111111111111111111111111111111111111111111

1222210000122221000012222100001222210000122221×9091＝11111111111111111111111111111111111111111111111111

1222222222222100000000000012222222222221×909090909091＝1111111111111111111111111111111111111111111111111111

1222222221000000001222222221000000001222222221×90909091＝111111111111111111111111111111111111111111111111111111

12222221000000122222210000001222222100000012222221×909091＝11111111111111111111111111111111111111111111111111111111

11222211000011222211000011222211000011222211000011222211×9091＝111111111111111111111111111111111111111111111111111111111111

我们注意到左边的对称数都是11的整数倍，因此等式右边都是偶数个1组成的纯元数．

前面已经知道9091、909091以及8个“90”和25个“90”且以91收尾的数是素数，由计算可以知道32个“90”且以“91”收尾的数也是素数．以下列出部分“9090…91”型数的分解式，作为建造等式队列的素材：

9090909090909091＝103×4013×21993833369

90909090909090909091＝7×7×13×127×2689×459691×909091

9090909090909090909091＝47×139×2531×549797184491917

909090909090909090909091＝251×5051×9091×78875943472201

90909090909090909090909091＝59×67×3767×6110428468208035141

9090909090909090909090909091＝311×643619×45457427387797819399

90909090909090909090909090909091＝6991×23561×552409422196840583651141

9090909090909090909090909090909091＝1681784709342383×5410329300988243277

909090909090909090909090909090909091＝11×5849×14142291752916723774213010912369

90909090909090909090909090909090909091＝97×73318244029×12794116754207799222751807

9099009090909090909090909090909090909091＝19×41×41×9293×30656095999733495085853303597733

909090909090909090909090909090909090909091＝57009401×2182600451×7306116556571817748755241

90909090909090909090909090909090909090909091＝7×13×19×211×241×2161×9091×29611×52579×3762091×8985695684401

9090909090909090909090909090909090909090909091＝6299×4855067598095567×297262705009139006771611927

909090909090909090909090909090909090909090909091＝197×909091×5076141624365532994918781726395939035533

90909090909090909090909090909090909090909090909091＝7×13×103×4013×21993833369×291078844423×377526955309799110357

909090909090909090909090909090909090909090909090909091＝11×23×331×4093×5171×8779×9091×20163494891×318727841165674579776721

909090909090909090909090909090909090909090909090909090909090909091（32个“90”，素数）．