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  内容 入力補助 youtubeの<IFRAME>タグが利用可能です。(詳細)
    
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怪しいがちょっと面白い話

 投稿者:壊れた扉  投稿日:2017年 7月23日(日)13時33分31秒
編集済
  [3]危うく間違われそうだった定理
 半径1の円に正三角形を外接させる→正三角形に円を外接させる→円に正方形を外接させる→正方形に円を外接させる→円に正五角形を外接させる→・・・この外接多角形の大きさは有限だろうか,無限だろうか?

円の直径Rは
R=1/cosπ/3・1/cosπ/4・1/cosπ/5・・・

=Π1/cosπ/n→8.7000・・・

 しかし,1965年になるまで数学者たちはR→12だとみなしていた.私もR=12と言及された記事を読んだことがある.

引用元:http://www.geocities.jp/ikuro_kotaro/koramu2/10602_n1.htm

解説
確かに円の半径は、R=(1/cos60°)(1/cos45°)(1/cos36°)(1/cos30°)・・・・で求まり、正n角形はnが大きくなれば円に近づくので(外接多角形の)大きさは無限大ではなく早い時期から収束するのは予測できるが、果たして1965年になるまでこんな計算ミスに気付かなかったのだろうか。
https://ja.wikipedia.org/wiki/%E8%A8%88%E7%AE%97%E6%A9%9F%E3%81%AE%E6%AD%B4%E5%8F%B2

https://ja.wikipedia.org/wiki/%E8%A8%88%E7%AE%97%E6%A9%9F%E3%81%AE%E6%AD%B4%E5%8F%B2_(1960%E5%B9%B4%E4%BB%A3%E4%BB%A5%E9%99%8D)

パソコンでプログラムを組める人は、本当にRが8.7に収束するかどうか確かめてみて下さい。(正しければ、多分、100とか1000とかでかなり近い値が出ると思うのですが。)

その後、電卓でちょっと手計算してみました。

n=10でR=5.4267455 n=20でR=6.8369245 n=30でR=7.3997746 n=40でR=7.7017366 n=50でR=7.8899571 n=60でR=8,0184897 n=70でR=8.1118304 以上から推定すると8.7は正しいですね。

おまけ:https://www.youtube.com/watch?v=rN_XZGkZ5nQ

 
 

Re:奇数の完全数はない

 投稿者:愛犬ベルのために  投稿日:2017年 7月23日(日)10時24分5秒
  (%i128) x:210*n+79;
(%o128)                           210 n + 79
(%i129) factor(1+x+x^2);
                                    2
(%o129)                   21 (2100 n  + 1590 n + 301)

2つの合成数から成る。21(=3x7)とn=301(=7x43)のとき、7、43を因数に持つ。

(%i130) factor(1+x+x^2+x^3+x^4);
                    4               3               2
(%o130) 1944810000 n  + 2935737000 n  + 1661864400 n  + 418117980 n + 39449441

n=39449441のとき、39449441を因数に持つ。

(%i131) factor(1+x+x^2+x^3+x^4+x^5+x^6);
                        6                    5                    4
(%o131) 85766121000000 n  + 193994797500000 n  + 182833532910000 n
                         3                   2
       + 91901598327000 n  + 25984570644900 n  + 3918406765770 n + 246203961361

n=246203961361(=281x337x1289x2017)のとき、281、337、1289、2017を因数に持つ。

(%i132) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8);
                  2                                  6                   5
(%o132) 63 (2100 n  + 1590 n + 301) (28588707000000 n  + 64528795800000 n
                   4                   3                  2
+ 60687796050000 n  + 30440230947000 n  + 8588496344400 n  + 1292364998190 n
+ 81029316187)

3つの合成数から成る。63(=3^2x7)とn=301(=7x43)のとき、7、43を因数に持ち、n=81029316187(=397x204104071)のとき、397、204104071を因数に持つ。

(%i134) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10);
                                  10                             9
(%o134) 166798809782010000000000 n   + 628275516845571000000000 n
                              8                              7
+ 1064929081310507700000000 n  + 1069666286380160490000000 n
                             6                             5
+ 705092508146426148000000 n  + 318704318143192787400000 n
                             4                            3
+ 100038728843726721300000 n  + 21532400227914465330000 n
                           2
+ 3041493599202575422500 n  + 254588311038284998950 n + 9589664237532319601

n=9589664237532319601(=5479x1750258119644519)のとき、5479、1750258119644519を因数に持つ。

(%i135) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12);
                                      12                                  11
(%o135) 7355827511386641000000000000 n   + 33241334801456772900000000000 n
                                  10                                  9
+ 68850545506578959760000000000 n   + 86427697647880931706000000000 n
                                  8                                  7
+ 73232329387809996740700000000 n  + 44125617143711803133070000000 n
                                  6                                 5
+ 19386799561341330684930000000 n  + 6257897522617546719950400000 n
                                 4                                3
+ 1472916614000734003414950000 n  + 246527811960743999096775000 n
                               2
+ 27852105905748744771387600 n  + 1907070708591259042808340 n
+ 59849094506439206629921

n=59849094506439206629921(=13x8346157x551604349222681)のとき、13、8346157、551604349222681を因数に持つ。

(%i136) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14);                  2                              4               3
(%o136) 21 (2100 n  + 1590 n + 301) (1944810000 n  + 2935737000 n
               2
+ 1661864400 n  + 418117980 n + 39449441)
                       8                         7                         6
(3782285936100000000 n  + 11364868693710000000 n  + 14940029447595000000 n
                         5                        4                        3
+ 11222751780752400000 n  + 5268975907112190000 n  + 1583186171980644000 n
                       2
+ 297314943935499000 n  + 31905269024495160 n + 1497907939519681)

4つの合成数から成る。21(=3x7)とn=301(=7x43)のとき、7、43を因数に持ち、n=39449441のとき、39449441を因数に持ち、n=1497907939519681(=631x201151x11801401)のとき、631、201151、11801401を因数に持つ。


(%i137) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16);
                                                16
(%o137) 14305686902419853283210000000000000000 n
                                           15
+ 86174733007433878110765000000000000000 n
                                            14
+ 243328380490397879067018600000000000000 n
                                            13
+ 427515052453624140455421720000000000000 n
                                            12
+ 523102678992420257431352484000000000000 n
                                            11
+ 472661978052907361595982014000000000000 n
                                            10
+ 326245774312730860031666345760000000000 n
                                            9
+ 175468821082004214511327441224000000000 n
                                           8
+ 74320033208812762225102479667800000000 n
                                           7
+ 24871793253541057820545832506860000000 n
                                          6
+ 6554795969935550679333531436260000000 n
                                          5
+ 1346083793699995809152981345822400000 n
                                         4
+ 211161880074987880408724629799160000 n
                                        3
+ 24461728402144071700110309850332000 n
                                       2
+ 1973493293934083822585469034790400 n  + 99067226245699570807359395172720 n
+ 2331127078802462119811160472961

n=2331127078802462119811160472961(=103x290194897x6860024417x11368765063)のとき、103、290194897、6860024417、11368765063を因数に持つ。

(%i138) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18);
                                                    18
(%o138) 630880792396715529789561000000000000000000 n
                                                17
+ 4274968417050124756621644300000000000000000 n
                                                 16
+ 13679341012771204846911216570000000000000000 n
                                                 15
+ 27464865986575621054817059365000000000000000 n
                                                 14
+ 38772390006617555496290800296600000000000000 n
                                                 13
+ 40869061563464787048537369191040000000000000 n
                                                 12
+ 33335046898578592011780239925180000000000000 n
                                                 11
+ 21512893206440990639520753259669200000000000 n
                                                 10
+ 11135668825495295949070493517050460000000000 n
                                                9
+ 4657885696722356403280166049609114000000000 n
                                                8
+ 1578143929682850532491184034523420600000000 n
                                               7
+ 432075287279981128581959591081187900000000 n
                                              6
+ 94883780834640598263441246922228848000000 n
                                              5
+ 16486022359904325278860104460413368400000 n
                                             4
+ 2216532496193632757216529679844829960000 n
                                            3
+ 222515019127949403786378855677384436000 n
                                           2
+ 15706424918450117475627769154815316100 n
+ 695625355474076714544064289765791710 n + 14548564098806166089741452511749681

n=14548564098806166089741452511749681(=200413027x1284297400723x56523439431961)のとき、200413027、1284297400723、56523439431961を因数に持つ。

(%i139) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20);
                   2                                  6                    5
(%o139) 147 (2100 n  + 1590 n + 301) (85766121000000 n  + 193994797500000 n
                    4                   3                   2
+ 182833532910000 n  + 91901598327000 n  + 25984570644900 n  + 3918406765770 n
                                                12
+ 246203961361) (1050832501626663000000000000 n
                                 11                                 10
+ 4738754185906904100000000000 n   + 9794354625195434910000000000 n
                                  9                                  8
+ 12268826995980830058000000000 n  + 10373696909598227751000000000 n
                                 7                                 6
+ 6237363531222775678860000000 n  + 2734606822997485875285000000 n
                                5                                4
+ 880835238805290258684600000 n  + 206881096313534053548690000 n
                               3                              2
+ 34552855045877533060500000 n  + 3895384929717877273836900 n
+ 266153645693635004462220 n + 8334805079410979166103)

4つの合成数から成る。147(=3x7^2)とn=301(=7x43)のとき、7、43を因数に持ち、n=246203961361(=281x337x1289x2017)のとき、281、337、1289、2017を因数に持ち、n=8334805079410979166103(=757x28393x387782571085603)のとき、757、28393、387782571085603を因数に持つ。

(%i140) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22);
                                                              22
(%o140) 1226943273861056329490036128410000000000000000000000 n
                                                          21
+ 10160258824973223604681775368119000000000000000000000 n
                                                          20
+ 40156128399209802564607331865173100000000000000000000 n
                                                           19
+ 100767038820353125459952949031280310000000000000000000 n
                                                           18
+ 180164900005681668455433553555063797000000000000000000 n
                                                           17
+ 244135492931224005850057023794177644500000000000000000 n
                                                           16
+ 260367629262553307900239199444591857170000000000000000 n
                                                           15
+ 224010040609624187405506667578803447057000000000000000 n
                                                           14
+ 158098397852004383559525168010862765494800000000000000 n
                                                          13
+ 92570341515939451414294832642102829215460000000000000 n
                                                          12
+ 45297501510380791393958597361994260185478000000000000 n
                                                          11
+ 18600387767634013567439893803350784438499800000000000 n
                                                         10
+ 6417897635786158263463384459032024852494340000000000 n
                                                         9
+ 1858270540129217143739317678355481796706682000000000 n
                                                        8
+ 449658786300855395106359286045106439688311400000000 n
                                                       7
+ 90269625401279928682075749394331821255030740000000 n
                                                       6
+ 14865505651328509045988900826571432331808705000000 n
                                                      5
+ 1974886607547887915615949183937265796441228300000 n
                                                     4
+ 206490451608484664309028427484317190887704150000 n
                                                    3
+ 16363147141522415990339509744912369046924475000 n
                                                  2
+ 923886725617476250082357318449595046691136100 n
+ 33119991916926567822790012399735120952863290 n
+ 566667750082192172494882844390303527173521

n=566667750082192172494882844390303527173521(=47x139x763539787x113601526281945630731567066551)のとき、47、139、763539787、113601526281945630731567066551を因数に持つ。

(%i141) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24);
                     4               3               2
(%o141) (1944810000 n  + 2935737000 n  + 1661864400 n  + 418117980 n
                                                               20
+ 39449441) (27821842944695154863719640100000000000000000000 n
                                                     19
+ 209326246917230212784176339800000000000000000000 n
                                                     18
+ 748092134816101308069163538190000000000000000000 n
                                                      17
+ 1688550818584914381070397700486000000000000000000 n
                                                      16
+ 2699671130189928587830409656848450000000000000000 n
                                                      15
+ 3249889817676325012811571338593197000000000000000 n
                                                      14
+ 3056443995195911718189933353135374500000000000000 n
                                                      13
+ 2299610244005266148839290334823690850000000000000 n
                                                      12
+ 1405773643211393449692973721424037515000000000000 n
                                                     11
+ 705118208341987524649274873211238690500000000000 n
                                                     10
+ 291784630024193100520171747378712769540000000000 n
                                                    9
+ 99787817194860272958761621875192346535000000000 n
                                                    8
+ 28154419851601757306806941249532159758000000000 n
                                                   7
+ 6517799760584957967169292300308750012950000000 n
                                                   6
+ 1225967097842687603985150316925434984965000000 n
                                                  5
+ 184478858536389512645005859895572357524800000 n
                                                 4
+ 21687246763082980548799648940184582176850000 n
                                                3
+ 1919655455853128532503121103213865400705000 n
                                               2
+ 120359350015843521195645619866183533412000 n
+ 4766109599930588890349325126902347389900 n
+ 89648252005978014962000699372137727201)

n=39449441のとき、39449441を因数に持ち、n=89648252005978014962000699372137727201(=101x2351x681251x12255372035351x45220376158151)のとき、101、2351、681251、12255372035351、45220376158151を因数に持つ。

(%i142) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24+x^25+x^26);
                   2                                  6                   5
(%o142) 189 (2100 n  + 1590 n + 301) (28588707000000 n  + 64528795800000 n
                   4                   3                  2
+ 60687796050000 n  + 30440230947000 n  + 8588496344400 n  + 1292364998190 n
                                                             18
+ 81029316187) (210293597465571843263187000000000000000000 n
                                                17
+ 1423988074266872195810723400000000000000000 n
                                                16
+ 4553371389858117521366194110000000000000000 n
                                                15
+ 9135653074255016741280745008000000000000000 n
                                                 14
+ 12887796301181184331449622422000000000000000 n
                                                 13
+ 13575145437244180829126935617840000000000000 n
                                                 12
+ 11064820923849026755169335618668000000000000 n
                                                11
+ 7135680432523045825782673582651200000000000 n
                                                10
+ 3691015652299123108693537704359460000000000 n
                                                9
+ 1542805484294342463422431313689587000000000 n
                                               8
+ 522349856825370234045258506367930300000000 n
                                               7
+ 142911302819754973125174347467302840000000 n
                                              6
+ 31361091452112896881134599809602000000000 n
                                             5
+ 5445112581795426052351848392613018600000 n
                                            4
+ 731571248234419486787079271998288180000 n
                                           3
+ 73389369664151288236143586150206876000 n
                                          2
+ 5176571610239242658087365844835543900 n
+ 229103169304705978090819401523741770 n + 4788135019860257234042655605668027)


4つの合成数から成る。189(=3^3x7)とn=301(=7x43)のとき、7、43を因数に持ち、n=81029316187(=397x204104071)のとき、397、204104071を因数に持ち、n=4788135019860257234042655605668027(=379x2355330006901x5363835045831949213)のとき、379、2355330006901、5363835045831949213を因数に持つ。
 

Re:次の問題をお願いできないでしょうか?

 投稿者:壊れた扉  投稿日:2017年 7月23日(日)07時56分32秒
  こんなのはどうでしょうか。

問題
x^3±3x+1=0は有理数解を持たない事を証明せよ。

ただし、あまり面白くないかもしれません。
 

次の問題をお願いできないでしょうか?

 投稿者:コルム  投稿日:2017年 7月22日(土)22時49分58秒
  お願いできないでしょうか?  

ちょっと考えてみました

 投稿者:壊れた扉  投稿日:2017年 7月22日(土)19時06分42秒
  問題
n1,n2,n3.mを整数とするとき
-16 n1 n3 - 8 n1 + 4 n2^2 + 4 n2 - 8 n3 - 3 = m^2
         に解が存在すれば 示し,
       存在しなければ 其の証明を願います;

解答
与式を変形すると、-8n3(2n1+1)+4n2(n2+1)-8(n1+n3)=m^2+3

ところで、連続する2数の積は偶数より、4n2(n2+1)は8の倍数。つまり、左辺は8の倍数。

よって、m^2=8p-3となる整数pを見つければ良い。ところが、平方数を8で割った余りは0か1か4しかない。https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1143373176(3番)

よって、余りが5となる事はないので、解は存在しない。

一応、平方数を8で割った余りは0か1か4しかない証明を自分でもしてみました。

(ⅰ)n=2m-1(mは整数)の場合、n^2=(2m-1)^2=4m^2-4m+1=4m(m-1)+1=8l+1

よって、n^2を8で割った余りは1

(ⅱ)n=2mの場合、n^2=4m^2 これを8で割った余りは0か4

(ⅰ),(ⅱ)より、平方数を8で割った余りは0か1か4のみ。

おまけ:https://www.youtube.com/watch?v=hAISLcT8acM
 

Re:奇数の完全数はない

 投稿者:愛犬ベルのために  投稿日:2017年 7月22日(土)12時58分40秒
  (%i114) x:210*n+73;
(%o114)                           210 n + 73
(%i115) factor(1+x+x^2);
                                   2
(%o115)                  3 (14700 n  + 10290 n + 1801)

2つの合成数から成る。3とn=1801のとき、1801を因数に持つ。

(%i116) factor(1+x+x^2+x^3+x^4);
                    4               3               2
(%o116) 1944810000 n  + 2713473000 n  + 1419755400 n  + 330162420 n + 28792661

n=28792661のとき、28792661を因数にもつ。

(%i117) factor(1+x+x^2+x^3+x^4+x^5+x^6);
                        6                    5                    4
(%o117) 85766121000000 n  + 179292033900000 n  + 156170187810000 n
                         3                   2
       + 72549960903000 n  + 18958412673900 n  + 2642218522650 n + 153436090543

n=153436090543のとき、153436090543を因数に持つ。

(%i118) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8);
                  2                                    6                   5
(%o118) 9 (14700 n  + 10290 n + 1801) (28588707000000 n  + 59627874600000 n
                   4                   3                  2
+ 51819462450000 n  + 24017912667000 n  + 6261815359800 n  + 870691188150 n
+ 50444871769)

3つの合成数から成る。9(=3^2)とn=1801のとき、1801を因数に持ち、n=50444871769(=19x181x14668471)のとき、19、181、14668471を因数に持つ。

(%i119) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10);
                                  10                             9
(%o119) 166798809782010000000000 n   + 580618714050711000000000 n
                             8                             7
+ 909499823052414300000000 n  + 844250689813597290000000 n
                             6                             5
+ 514292921987081202000000 n  + 214829478167039530200000 n
                            4                            3
+ 62318317196806736940000 n  + 12396005854405604802000 n
                           2
+ 1618147708585358750100 n  + 125173326394085680830 n + 4357315077338329283

n=4357315077338329283(=23x3323x461561x123518407)のとき、23、3323、461561、123518407を因数に持つ。

(%i120) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12);
                                      12                                  11
(%o120) 7355827511386641000000000000 n   + 30719336797552781700000000000 n
                                  10                                  9
+ 58799582826734601180000000000 n   + 68210837122742901846000000000 n
                                  8                                  7
+ 53411768566361489724300000000 n  + 29741212901307012893550000000 n
                                  6                                 5
+ 12075576570249764820444000000 n  + 3602169752585535382784400000 n
                                4                                3
+ 783516165786473265261990000 n  + 121190867637333745792527000 n
                               2
+ 12653080921204664074911000 n  + 800643936625275768960060 n
+ 23220132047135956749181

n=23220132047135956749181(=147083x157870943937341207)のとき、147083、157870943937341207を因数に持つ。

(%i121) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14);                  2                                4               3
(%o121) 3 (14700 n  + 10290 n + 1801) (1944810000 n  + 2713473000 n
               2
+ 1419755400 n  + 330162420 n + 28792661)
                       8                         7                         6
(3782285936100000000 n  + 10500346194030000000 n  + 12753507958821000000 n
                        5                        4                        3
+ 8851490067344400000 n  + 3839563778143590000 n  + 1065923724252492000 n
                       2
+ 184947801175481400 n  + 18337197694173960 n + 795414738437281)

3つの合成数から成る。3とn=1801のとき、1801を因数に持ち、n=28792661のとき、28792661を因数に持ち、n=795414738437281(=61x13039585876021)のとき、61、13039585876021を因数に持つ。

(%i122) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16);
                                                16
(%o122) 14305686902419853283210000000000000000 n
                                           15
+ 79634990423470516609869000000000000000 n
                                            14
+ 207797725469489794484025600000000000000 n
                                            13
+ 337383256155703225195225680000000000000 n
                                            12
+ 381490039312851656286307404000000000000 n
                                            11
+ 318545243034981910826270943600000000000 n
                                            10
+ 203184044671662483109340451120000000000 n
                                            9
+ 100987813204548757598857038552000000000 n
                                           8
+ 39527492253338114889382003224600000000 n
                                           7
+ 12224324706945788527581923249580000000 n
                                          6
+ 2977154831378660622496586720136000000 n
                                         5
+ 564987524259523166714637266220000000 n
                                        4
+ 81904328877740927620485466685880000 n
                                       3                                      2
+ 8768051738723799390002189160924000 n  + 653696713461475196003170212775200 n
+ 30324621398681086307212868507280 n + 659410905926390259528818985041

n=659410905926390259528818985041(=137x155072369x909139159x34140570383)のとき、137、155072369、909139159、34140570383を因数に持つ。

(%i123) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18);
                                                    18
(%o123) 630880792396715529789561000000000000000000 n
                                                17
+ 3950515438103242484158441500000000000000000 n
                                                 16
+ 11681723505091101374150338590000000000000000 n
                                                 15
+ 21674054723327743713003669285000000000000000 n
                                                 14
+ 28275235446457530041517148287600000000000000 n
                                                 13
+ 27542245195228476536242091268120000000000000 n
                                                 12
+ 20759973940971047367405113181084000000000000 n
                                                 11
+ 12380712972087190545035171489926800000000000 n
                                                10
+ 5922216535278965147092378408227660000000000 n
                                                9
+ 2289169688630696205819124192616322000000000 n
                                               8
+ 716732329796795623953277936374013800000000 n
                                               7
+ 181338943313228813401344921276683580000000 n
                                              6
+ 36799756493722237656818498294757252000000 n
                                             5
+ 5908676321848255349365482942272209200000 n
                                            4
+ 734124659962404148730773978026370680000 n
                                           3
+ 68104604754069792364926952263422676000 n
                                          2
+ 4442382699071117236125261529552553700 n
+ 181817445809274634288290966356652390 n + 3514000717681733693029076371283563

n=3514000717681733693029076371283563(=48527x67679x1069952429562890572890811)のとき、48527、67679、1069952429562890572890811を因数に持つ。

(%i124) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20);
                  2                                    6                    5
(%o124) 3 (14700 n  + 10290 n + 1801) (85766121000000 n  + 179292033900000 n
                    4                   3                   2
+ 156170187810000 n  + 72549960903000 n  + 18958412673900 n  + 2642218522650 n
                                                12
+ 153436090543) (7355827511386641000000000000 n
                                  11                                  10
+ 30649281297444337500000000000 n   + 58531537139414911110000000000 n
                                  9                                  8
+ 67744658277803581326000000000 n  + 52925308283476752870600000000 n
                                  7                                  6
+ 29402796968811410698260000000 n  + 11910778505550476579727000000 n
                                 5                                4
+ 3544847237495824556095800000 n  + 769274191466224537860270000 n
                                3                               2
+ 118713941402309687823732000 n  + 12365894058375441402963900 n
+ 780665237618207161460940 n + 22588379426199821593969)

4つの合成数から成る。3とn=1801のとき、1801を因数に持ち、n=153436090543のとき、153436090543を因数に持ち、n=22588379426199821593969(=127x211x40226467x20954968231)のとき、127、211、40226467、20954968231を因数に持つ。


(%i125) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22);
                                                              22
(%o125) 1226943273861056329490036128410000000000000000000000 n
                                                         21
+ 9389037338546273911859466944547000000000000000000000 n
                                                          20
+ 34291339550153953309644959171373300000000000000000000 n
                                                          19
+ 79518368591575679487104360404369110000000000000000000 n
                                                           18
+ 131381896209018082807687093485363303000000000000000000 n
                                                           17
+ 164517713906703145466821157820454514100000000000000000 n
                                                           16
+ 162138130787386014291890685532758263310000000000000000 n
                                                           15
+ 128908666261598551621986801080266433937000000000000000 n
                                                          14
+ 84073394660821842957869155898133679077000000000000000 n
                                                          13
+ 45490372398276080141931511870309937280300000000000000 n
                                                          12
+ 20570195215919690464428889910256972479634000000000000 n
                                                         11
+ 7805535975053144091012403305330261342283800000000000 n
                                                         10
+ 2488800648744559032656481860826077148182220000000000 n
                                                        9
+ 665921273005820765714072163139895438717514000000000 n
                                                        8
+ 148906508227946560502165781860886814376626200000000 n
                                                       7
+ 27624149315765461121588939081582824054448340000000 n
                                                      6
+ 4203815657007075926037327298802961334600029000000 n
                                                     5
+ 516086945847636877657638028864649675845806300000 n
                                                    4
+ 49865186012280087915293441237361564693939450000 n
                                                   3
+ 3651582297540687704075032665876031877005995000 n
                                                  2
+ 190524048107786368064308881502149400472537100 n
+ 6311580698536152896397067791010305502993770 n
+ 99791439254898834712459730799116101807103

n=99791439254898834712459730799116101807103(=124247x803169808968416418202932310632177049)のとき、124247、803169808968416418202932310632177049を因数に持つ。

(%i126) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24);
                     4               3               2
(%o126) (1944810000 n  + 2713473000 n  + 1419755400 n  + 330162420 n
                                                               20
+ 28792661) (27821842944695154863719640100000000000000000000 n
                                                     19
+ 193428050948832981433479402600000000000000000000 n
                                                     18
+ 638773111109598441067228408110000000000000000000 n
                                                      17
+ 1332298203171448177083076394058000000000000000000 n
                                                      16
+ 1968311988256865699714402148840450000000000000000 n
                                                      15
+ 2189512764080086348286603627725773000000000000000 n
                                                      14
+ 1902790854498466286561962407000253500000000000000 n
                                                      13
+ 1322892689318598710514045546763070850000000000000 n
                                                     12
+ 747276882240903141633879813650351505000000000000 n
                                                     11
+ 346356904150506266566039283265407434500000000000 n
                                                     10
+ 132440282873308022277188398618401492240000000000 n
                                                    9
+ 41853422726470366468474683978129243195000000000 n
                                                    8
+ 10911785210941808591253053333976433257000000000 n
                                                   7
+ 2334242697175149603844241366217998263350000000 n
                                                  6
+ 405713611660896663555517496646403639815000000 n
                                                 5
+ 56413511718414961254623738029408264351200000 n
                                                4
+ 6128253505385697109082579429546354521050000 n
                                               3
+ 501246505230798979360599661307454783705000 n
                                              2
+ 29040472130601468408602776281747206199000 n
+ 1062633817411398063120824781115470764100 n
+ 18469587781044478319759952212422801301)

2つの合成数から成る。n=28792661のとき、28792661を因数に持ち、n=18469587781044478319759952212422801301(=1901x2246201x4325401928085456016477853201)1901、2246201、4325401928085456016477853201を因数に持つ。

(%i127) factor(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24+x^25+x^26);
                   2                                    6                   5
(%o127) 27 (14700 n  + 10290 n + 1801) (28588707000000 n  + 59627874600000 n
                   4                   3                  2
+ 51819462450000 n  + 24017912667000 n  + 6261815359800 n  + 870691188150 n
                                                             18
+ 50444871769) (210293597465571843263187000000000000000000 n
                                                17
+ 1315837081284578104989655800000000000000000 n
                                                16
+ 3887985280652765305457530590000000000000000 n
                                                15
+ 7208201282226079169483167824000000000000000 n
                                                14
+ 9396405242901853203076272342000000000000000 n
                                                13
+ 9145834436424470450994238412880000000000000 n
                                                12
+ 6888410222354620998407565280812000000000000 n
                                                11
+ 4104930173321529248030630738769600000000000 n
                                                10
+ 1962058886414992848909878859066660000000000 n
                                               9
+ 757832268297854380796143896057147000000000 n
                                               8
+ 237093238224614441992681936038741300000000 n
                                              7
+ 59940455031677417370520812656277240000000 n
                                              6
+ 12154592270312365189722929960699304000000 n
                                             5
+ 1950077441170994854952125552703337000000 n
                                            4
+ 242101451029732354550192035065693660000 n
                                           3
+ 22442420222438681778674217787212156000 n
                                          2
+ 1462764889498235512077121763517277500 n
+ 59821757385642124852524316216191810 n + 1155287907183035754318353097881161)

4つの合成数から成る。27(=3^3)とn=1801のとき、1801を因数に持ち、n=50444871769(=19x181x14668471)のとき、19、181、14668471を因数に持ち、n=1155287907183035754318353097881161(=109x20359x13909795579x37427131197853489)のとき、109、20359、13909795579、37427131197853489を因数に持つ。
 

(無題)

 投稿者:  投稿日:2017年 7月22日(土)09時18分44秒
        n1,n2,n3.mを整数とするとき
 -16 n1 n3 - 8 n1 + 4 n2^2 + 4 n2 - 8 n3 - 3 = m^2
         に解が存在すれば 示し,
       存在しなければ 其の証明を願います;
 
 

Re:次の文章を解説して下さい

 投稿者:壊れた扉  投稿日:2017年 7月22日(土)07時57分46秒
  問題
a,b,cがすべて奇数のとき、ax^2+bx+c=0は有理数の解を持たないことを証明せよ。

解答
有理数v/u(u,vは互いに素な整数)が解ならば、a(v/u)^2+b(v/u)+c=0 ∴av^2+buv+cu^2=0

法2の剰余系では、a,b,c≡1(2)だから、v^2+uv+u^2≡0(2)―――①

一方、u,vは互いに素だから、少なくとも一方は奇数で、u≡0,v≡1;u≡1,v≡0;u≡1,v≡1(2)のいずれかで、いずれの場合もu^2+uv+v^2≡1(2)

これは①と矛盾するから、有理数の解は存在しない。

解説
有理数v/u(u,vは互いに素な整数)が解ならば、a(v/u)^2+b(v/u)+c=0 ∴av^2+buv+cu^2=0

ここで、a,b,cが奇数より、v^2,uv,u^2のうち2つが奇数で残りが偶数か、3つとも偶数でなければ0とならない。

よって、v^2+uv+u^2は偶数。∴v^2+uv+u^2≡0(mod2)―――①

また、u,vは互いに素より偶数同士ではないので、少なくとも一方は奇数。よって、u≡0,v≡1(mod2)またはu≡1,v≡0(mod2)またはu≡1,v≡1(mod2)と置ける。

∴v^2+uv+u^2≡1(mod2) これは①と矛盾するので、背理法により、有理数解は存在しない。

別解2
ax^2+bx+c=0が有理数解を持つと仮定すると、判別式D=b^2-4ac=d^2(dは0以上の整数)となるdが存在する。

∴b^2-d^2=4ac ∴(b+d)(b-d)=4ac―――① bは奇数よりdも奇数と分かる。

よって、b=2n-1,d=2m-1と置いて①に代入すると、(2n-1+2m-1)(2n-1-2m+1)=4ac

∴(n+m-1)(n-m)=ac ところで、n,mは整数より、n+mとn-mの偶奇は一致している。よって、n+m-1とn-mの偶奇は一致していない。

ところが、aとcは奇数同士で偶奇が一致しているので矛盾が生じる。よって、背理法により有理数解を持たない。

おまけ:https://www.youtube.com/watch?v=xBMfLT_3vJY
 

次の文章を解説して下さい

 投稿者:壊れた扉  投稿日:2017年 7月21日(金)22時28分0秒
  問題
a,b,cがすべて奇数のとき、ax^2+bx+c=0は有理数の解を持たないことを証明せよ。

解答
有理数v/u(u,vは互いに素な整数)が解ならば、a(v/u)^2+b(v/u)+c=0 ∴av^2+buv+cu^2=0

法2の剰余系では、a,b,c≡1(2)だから、v^2+uv+u^2≡0(2)―――①

一方、u,vは互いに素だから、少なくとも一方は奇数で、u≡0,v≡1;u≡1,v≡0;u≡1,v≡1(2)のいずれかで、いずれの場合もu^2+uv+v^2≡1(2)

これは①と矛盾するから、有理数の解は存在しない。

因みに、単純に考えると簡単に解けますよね。

別解
ax^2+bx+c=0が有理数解を持つと仮定すると、因数分解出来、(px+q)(mx+n)=0(p,q,m,nは整数)と置ける。

∴pmx^2+(pn+qm)x+qn=0 ところで、a,b,cは奇数より、pm,qnは奇数。

よって、p,m,q,nは全て奇数。よって、pn+qmは偶数。ところが、b=pn+qmは奇数より矛盾。

よって、有理数解を持たない。よって、示された。

おまけ:https://www.youtube.com/watch?v=42yiDEEWQnQ
 

Re:奇数の完全数はない

 投稿者:愛犬ベルのために  投稿日:2017年 7月21日(金)20時50分32秒
  1が並んだ数の話で、
==============================
\newpage
\subsection{素数探求$x^5+x^3+x^2+x+1$}
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
>                               5    3    2
> (%o19)                       x  + x  + x  + x + 1
\end{verbatim}
\end{quote}
では、どうなんだろう。

\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i1) factor(2^5+2^3+2^2+2+1);
(%o1)                                 47
(%i2) factor(3^5+3^3+3^2+3+1);
(%o2)                                 283
(%i3) factor(4^5+4^3+4^2+4+1);
(%o3)                                1109
(%i4) factor(5^5+5^3+5^2+5+1);
(%o4)                               17 193
(%i5) factor(6^5+6^3+6^2+6+1);
(%o5)                               5 1607
(%i6) factor(7^5+7^3+7^2+7+1);
(%o6)                                17207
(%i7) factor(8^5+8^3+8^2+8+1);
(%o7)                                33353
(%i8) factor(9^5+9^3+9^2+9+1);
(%o8)                              19 23 137
(%i9) factor(10^5+10^3+10^2+10+1);
(%o9)                               101111
(%i10) factor(11^5+11^3+11^2+11+1);
(%o10)                              5 32503
(%i11) factor(12^5+12^3+12^2+12+1);
(%o11)                              409 613
(%i12) factor(13^5+13^3+13^2+13+1);
(%o12)                             19 71 277
(%i13) factor(14^5+14^3+14^2+14+1);
(%o13)                              540779
(%i14) factor(15^5+15^3+15^2+15+1);
(%o14)                             509 1499
(%i15) factor(16^5+16^3+16^2+16+1);
(%o15)                             5 251 839
(%i16) factor(17^5+17^3+17^2+17+1);
(%o16)                              1425077
(%i17) factor(18^5+18^3+18^2+18+1);
(%o17)                             31 61153
(%i18) factor(19^5+19^3+19^2+19+1);
(%o18)                             47 52837
(%i19) factor(20^5+20^3+20^2+20+1);
(%o19)                              3208421
(%i20) factor(21^5+21^3+21^2+21+1);
                                    2
(%o20)                             5  163753
(%i21) factor(22^5+22^3+22^2+22+1);
(%o21)                            17 149 2039
(%i22) factor(23^5+23^3+23^2+23+1);
(%o22)                             37 174299
(%i23) factor(24^5+24^3+24^2+24+1);
(%o23)                              7977049
(%i24) factor(25^5+25^3+25^2+25+1);
(%o24)                            89 131 839
(%i25) factor(26^5+26^3+26^2+26+1);
(%o25)                             5 2379931
\end{verbatim}
\end{quote}
かなり、素数がえられる。
これは、6から5つおきに5の倍数になっている。
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
                              5    3    2
(%o19)                       x  + x  + x  + x + 1
(%i5) factor(6^5+6^3+6^2+6+1);
(%o5)                               5 1607
(%i10) factor(11^5+11^3+11^2+11+1);
(%o10)                              5 32503
(%i15) factor(16^5+16^3+16^2+16+1);
(%o15)                             5 251 839
(%i20) factor(21^5+21^3+21^2+21+1);
                                    2
(%o20)                             5  163753
(%i25) factor(26^5+26^3+26^2+26+1);
(%o25)                             5 2379931
\end{verbatim}
\end{quote}
なぜなんだろうか?
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i1) factor((5*x+1)^5+(5*x+1)^3+(5*x+1)^2+(5*x+1)+1);
                        5        4        3       2
(%o1)           5 (625 x  + 625 x  + 275 x  + 70 x  + 11 x + 1)
(%i2) factor((5*x+1)^7+(5*x+1)^5+(5*x+1)^3+(5*x+1)^2+1);
                7          6          5         4         3        2
(%o2) 5 (15625 x  + 21875 x  + 13750 x  + 5000 x  + 1150 x  + 175 x  + 17 x
                                                                           + 1)\end{verbatim}
\end{quote}
なるほどこうなるわけだ。

\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
> (%i4) factor(5^5+5^3+5^2+5+1);
> (%o4)                               17 193
> (%i21) factor(22^5+22^3+22^2+22+1);
> (%o21)                            17 149 2039
\end{verbatim}
\end{quote}
さて、これも、
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i3) factor((17*x+5)^5+(17*x+5)^3+(17*x+5)^2+(17*x+5)+1);
                   5           4          3          2
(%o3)   17 (83521 x  + 122825 x  + 72539 x  + 21522 x  + 3211 x + 193)
\end{verbatim}
\end{quote}
より、5から17おきに現れるんだな。

\newpage
\subsection{素数探求$x^5+x^3+1$}
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i1) factor(x^5+x^3+1);
                                   5    3
(%o1)                             x  + x  + 1
(%i2) factor(2^5+2^3+1);
(%o2)                                 41
(%i3) factor(3^5+3^3+1);
(%o3)                                 271
(%i4) factor(4^5+4^3+1);
                                     2   2
(%o4)                               3  11
(%i5) factor(5^5+5^3+1);
(%o5)                                3251
(%i6) factor(6^5+6^3+1);
(%o6)                                7993
(%i7) factor(7^5+7^3+1);
(%o7)                               3 5717
(%i8) factor(8^5+8^3+1);
(%o8)                               23 1447
(%i9) factor(9^5+9^3+1);
(%o9)                                59779
(%i10) factor(10^5+10^3+1);
(%o10)                             3 131 257
(%i11) factor(11^5+11^3+1);
(%o11)                             13 12491
(%i12) factor(12^5+12^3+1);
(%o12)                              43 5827
(%i13) factor(13^5+13^3+1);
                                    5
(%o13)                             3  29 53
(%i14) factor(14^5+14^3+1);
(%o14)                            19 23 1237
(%i15) factor(15^5+15^3+1);
(%o15)                             11 69341
(%i16) factor(16^5+16^3+1);
(%o16)                             3 350891
(%i17) factor(17^5+17^3+1);
(%o17)                              1424771
(%i18) factor(18^5+18^3+1);
(%o18)                             109 17389
(%i19) factor(19^5+19^3+1);
(%o19)                            3 37 22369
(%i20) factor(20^5+20^3+1);
(%o20)                              3208001
(%i21) factor(21^5+21^3+1);
(%o21)                             71 57653
(%i22) factor(22^5+22^3+1);
                                    2
(%o22)                             3  573809
(%i23) factor(23^5+23^3+1);
(%o23)                              6448511
(%i24) factor(24^5+24^3+1);
(%o24)                             13 613573
(%i25) factor(25^5+25^3+1);
(%o25)                             3 3260417
(%i26) factor(26^5+26^3+1);
(%o26)                            11 1081723\end{verbatim}
\end{quote}
において
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
> (%i4) factor(4^5+4^3+1);
>                                      2   2
> (%o4)                               3  11
> (%i7) factor(7^5+7^3+1);
> (%o7)                               3 5717
> (%i10) factor(10^5+10^3+1);
> (%o10)                             3 131 257
> (%i13) factor(13^5+13^3+1);
>                                     5
> (%o13)                             3  29 53
> (%i16) factor(16^5+16^3+1);
> (%o16)                             3 350891
> (%i19) factor(19^5+19^3+1);
> (%o19)                            3 37 22369
> (%i22) factor(22^5+22^3+1);
>                                     2
> (%o22)                             3  573809
> (%i25) factor(25^5+25^3+1);
> (%o25)                             3 3260417

\end{verbatim}
\end{quote}
3が定期的に出てくるのでxを3x+1とおいてみると
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i27) factor((3*x+1)^5+(3*x+1)^3+1);
                        5        4       3       2
(%o27)           3 (81 x  + 135 x  + 99 x  + 39 x  + 8 x + 1)
\end{verbatim}
\end{quote}
となり、3が頭にある。
また、11も定期的に現れるので
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i28) factor((11*x+4)^5+(11*x+4)^3+1);
                     5          4          3         2
(%o28)    11 (14641 x  + 26620 x  + 19481 x  + 7172 x  + 1328 x + 99)
\end{verbatim}
\end{quote}
となり11が頭にある。
また、最初の41は
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i29) factor((41*x+2)^5+(41*x+2)^3+1);
                       5           4          3         2
(%o29)    41 (2825761 x  + 689210 x  + 68921 x  + 3526 x  + 92 x + 1)
\end{verbatim}
\end{quote}
となり周期的に現れる。
次の271は
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i30) factor((271*x+3)^5+(271*x+3)^3+1);
                        5              4            3          2
(%o30) 271 (5393580481 x  + 298537665 x  + 6683131 x  + 75609 x  + 432 x + 1)
\end{verbatim}
\end{quote}
となり周期的に現れる。

そこで、$x^5+x^3+1$をいれてみると
\begin{quote}
\setlength{\baselineskip}{12pt}
\begin{verbatim}
(%i1) factor(((x^5+x^3+1)*x+x)^5+((x^5+x^3+1)*x+x)^3+1);
        5    3        25      23      21      20      19       18    17
(%o1) (x  + x  + 1) (x   + 4 x   + 6 x   + 9 x   + 4 x   + 27 x   + x
       16       15      14       13       11       10    9       8      6
+ 27 x   + 31 x   + 9 x   + 63 x   + 33 x   + 49 x   + x  + 54 x  + 5 x
       5      3
+ 31 x  + 7 x  + 1)
\end{verbatim}
\end{quote}
となり、$x^5+x^3+1$を因数にもつので、全ての数が繰り返して現れることがわかった。
=======================
これを応用できないだろうか?
 

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