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电算游戏(六)之二“9090…91”型数等式队列

(2010-10-04 07:36:01)
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分类: 科普

6·2 “9090…91”型数等式队列

我们以出现“90” 的个数作为“909091”的标志,作为特例,我们不妨将91归入这一类型的数。它们不是回文数,却是回文数的再造者,我们有等式队列:

11×9091100001

121×90911100011

12321×9091112010211

1234321×909111221212211

123454321×90911122323232211

12345654321×9091112234343432211

1234567654321×909111223454545432211

123456787654321×90911122345656565432211

12345678987654321×9091112234567676765432211

12345678987654321×90909111223345656565654332211

12345678987654321×909090911122334454545454544332211

12345678987654321×9090909091112233445343434343544332211

12345678987654321×90909090909111223344534232323243544332211

12345678987654321×909090909090911122334453423121213243544332211

12345678987654321×9090909090909091112233445342312010213243544332211

123456789987654321×909090909090909091112233445443322110011223344544332211

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回文数等式队列,如:

123456789987654321×90911122345677777765432211

123456789987654321×909091112233456666666654332211

123456789987654321×9090909111223344555555555544332211

123456789987654321×90909090911122334454444444444544332211

123456789987654321×909090909091112233445443333333344544332211

123456789987654321×9090909090909111223344544332222223344544332211

123456789987654321×90909090909090911122334454433221111223344544332211

  1234567899987654321×909111223456778787765432211

                  1234567899987654321×9090911122334566767676654332211

 1234567899987654321×90909091112233445565656565544332211

 1234567899987654321×909090909111223344545454545454544332211

1234567899987654321×9090909090911122334454534343434354544332211

1234567899987654321×90909090909091112233445453423232324354544332211

1234567899987654321×909090909090909111223344545342312121324354544332211

1234567899987654321×9090909090909090911122334454534231201021324354544332211

 

在纯元数的分解中,对出现“909091”型的分解式进行变换,可以看到回文数经过“909091”型的数的整合,又会得到纯元数等式队列:

1001×91111111
122221
×90911111111111

11222211×9901111111111111

12222221×90909111111111111111
1222222221×90909091111111111111111111

12222100001222211×909111111111111111111111

122222222221×90909090911111111111111111111111

1222222100000012222221×9090911111111111111111111111111111

12222100001222210000122221×9091111111111111111111111111111111

122222222222222221×90909090909090911111111111111111111111111111111111

11222211000011222211000011222211×9901111111111111111111111111111111111111

12222222222222222221×90909090909090909111111111111111111111111111111111111111

122221000012222100001222210000122221×90911111111111111111111111111111111111111111

122222210000001222222100000012222221×909091111111111111111111111111111111111111111111

1222222222210000000000122222222221×9090909091

11111111111111111111111111111111111111111111

122222222222222222222221×9090909090909090909091

1111111111111111111111111111111111111111111111

11222211000011222211000011222211000011222211×9901

111111111111111111111111111111111111111111111111

1222210000122221000012222100001222210000122221×9091

11111111111111111111111111111111111111111111111111

1222222222222100000000000012222222222221×909090909091

1111111111111111111111111111111111111111111111111111

1222222221000000001222222221000000001222222221×90909091

111111111111111111111111111111111111111111111111111111

12222221000000122222210000001222222100000012222221×909091

11111111111111111111111111111111111111111111111111111111

11222211000011222211000011222211000011222211000011222211×9091

111111111111111111111111111111111111111111111111111111111111

我们注意到左边的回文数都是11的整数倍,因此等式右边都是偶数个1组成的纯元数。

在50个“90” 而以“91” 为末两位的100个数的分解中中,前面已经知道9091,909091,以及8个“90” 和25个“90”的数是质数,由计算可以知道32个90的也是质数,还有7个数有待进行探究。列出分解式,提供建造等式队列的素材:

91=7×13

90909091=7×13×19×52579

9090909091=11×23×4093×8779

909090909091=859×1058313049

90909090909091=7×13×211×241×2161×9091

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

=57009401×2182600451×730611655657181774875524

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

90909090909090909090909090909090909090909090909090909091

=7×12987012987012987012987012987012987012987012987012987013  (待进一步分解)

909090909090909090909090909090909090909090909090909090909090909091 (32个“90”,质数)

您找出一个质因数,说明它不是质数,就是成绩。如果进一步分解出来,就是成就!

 

6·3 “99…90…01”型数等式队列

“99…90…01”型的自然数也是从纯元数分解中得到,或循环节的质因数,同回文数亦有密切关系。

1111222222221111 × 99990001 990000999901×112233332211

1111222222221111×99990001×2990000999901×224466664422

1111222222221111×99990001×3990000999901×336699996633

1111222222221111×99990001×4990000999901×448933328844

1111222222221111×99990001×5990000999901×561166661055

1111222222221111×99990001×6990000999901×673399993266

1111222222221111×99990001×7990000999901×785633325477

1111222222221111×99990001×8990000999901×897866657688

1111222222221111×99990001×9990000999901×1010099989899

2222444444442222×8999100091980001999802×1010099989899

3333666666663333×8999100092970002999703×1010099989899

4444888888884444×8999100093960003999604×1010099989899

5556111111105555×8999100094950004999505×1010099989899

6667333333326666×8999100095940005999406×1010099989899

7778555555547777×8999100096930006999307×1010099989899

8889777777768888×8999100097920007999208×1010099989899

10000999999989999×8999100098910008999109×1010099989899

97的循环节有两个质因数9999999900000001与99990001,由此,我们追寻由它们引出的等式队列,从而构建数字大厦。

999001×1110989011

99990001×11111098890111

9999900001×111111109888901111

999999000001×1111111110988889011111

99999990000001×11111111111098888890111111

9999999900000001×111111111111109888888901111111

999999999000000001×1111111111111110988888889011111111

99999999990000000001×11111111111111111098888888890111111111

9999999999900000000001×111111111111111111109888888888901111111111

999999999999000000000001×1111111111111111111110988888888889011111111111

99999999999990000000000001×11111111111111111111111098888888888890111111111111

9999999900000001×99990001999900000000999999990001

9999999900000001×99990002999900010000999899990002

9999999900000001×99990003999900020000999799990003

9999999900000001×99990004999900030000999699990004

9999999900000001×99990005999900040000999599990005

9999999900000001×99990006999900050000999499990006

9999999900000001×99990007999900060000999399990007

9999999900000001×99990008999900070000999299990008

9999999900000001×99990009999900080000999199990009

9999999900000002×99990002999900010000999999980004

9999999900000003×99990003999900020000999999970009

9999999900000004×99990004999900030000999999960016

9999999900000005×99990005999900040000999999950025

9999999900000006×99990006999900050000999999940036

9999999900000007×99990007999900060000999999930049

9999999900000008×99990008999900070000999999920064

9999999900000009×99990009999900080000999999910081

999999999000000001×10101011010100998989899001010101

999999999000000001×20202022020201997979798002020202

999999999000000001×30303033030302996969697003030303

999999999000000001×40404044040403995959596004040404

999999999000000001×50505055050504994949495005050505

999999999000000001×60606066060605993939394006060606

999999999000000001×70707077070706992929293007070707

999999999000000001×80808088080807991919192008080808

999999999000000001×90909099090908990909091009090909

 

9999900001×1011009989900101

99999990000001×101011010099898990010101

999999999000000001×10101011010100998989899001010101

9999999999900000000001×1010101011010101009989898989900101010101

99999999999990000000000001×101010101011010101010099898989898990010101010101

999999999999999000000000000001×10101010101011010101010100998989898989899001010101010101

9999999999999999900000000000000001×1010101010101011010101010101009989898989898989900101010101010101

 

以上都可以通过分解质因数得出的奥秘。综合起来,也能够得到不同结构的数字大厦。

“99…90…01”型的自然数,有的是质数,有的不是质数是合数,提供构建等式队列的素材如下:

99900119×52579

99999000017×13×211×241×2161

9999999000000172×13×127×2689×459691

99999999900000000170541929×14175966169

9999999999000000000161×9901×4188901×39526741

99999999999000000000017×13×599144041×183411838171

9999999999990000000000013169×98641×3199044596370769

999999999999900000000000017×13×13×157×6397×216451×388847808493

99999999999999000000000000019901×226549×4458192223320340849

99999999999999900000000000000119×29611×52579×3762091×8985695684401

9999999999999999000000000000000197×206209×66554101249×75118313082913

99999999999999999000000000000000017×13×291078844423×377526955309799110357

999999999999999999000000000000000001109×153469×59779577156334533866654838281

999999999999999999900000000000000000017×13×1458973×753201806271328462547977919407

999999999999999999990000000000000000000199990001×100009999999899989999000000010001

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