Computing Minimal Equal Sums Of Like Powers ①
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$${\rm Published}$$ $${\rm Online}$$ $${\rm First}$$ $${\rm (4/24/2024)}$$
$${\rm Latest}$$ $${\rm Additions}$$ $${\rm (4/24/2024)}$$
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$${\small \rm \bf Introduction}$$
$${\small \rm Search~for~counterexamples~Euler's~conjecture.}$$
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\small \rm Counterexamples~have~been~found~for~small}$$$${\small k,}$$
$${\small \rm so~we~will~search~for}$$ $${\small 11≦k≦20.}$$
オイラー予想の反例を探す.
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
小さな$${\small k}$$に対しては反例が見つかっているので,
$${\small 11≦k≦20}$$の範囲で検索します.
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Equal Sums Of Like Powers
こちらも参照して下さい.過去の投稿.
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$${\small 21≦k≦32}$$
$${\footnotesize \rm Computing~Minimal~Equal~Sums~Of~Like~Powers~②}$$
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$${\small \rm \bf Latest~update}$$
$${\footnotesize (4/23/24)…}$$
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}~~,}$$$${\footnotesize ~~(k,b_k)~~,~b≦b_k}$$
$${\footnotesize (11,80)~,~\rm Complete}$$
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$${\small \rm \bf Recent~update~history}$$
$${\footnotesize (4/15/24)…}$$$${\footnotesize (13,1,n)→n≦6000,b=19}$$
$${\footnotesize (4/10/24)…}$$$${\footnotesize (13,1,n)→n≦6000,b=18}$$
$${\footnotesize (4/8/24)…}$$$${\footnotesize (13,1,n)→n≦6000,b=17}$$
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}~~,}$$$${\footnotesize ~~(k,b_k)~~,~b≦b_k}$$
$${\footnotesize (4/7/24)…}$$$${\footnotesize (19,24),(20,20)~,~\rm Complete}$$
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Abstract(要約)
$${\small \rm In~number~theory,Euler's~conjecture~is}$$
$${\small \rm a~disproved~conjecture~related~to}$$
$${\small \rm Fermat's~Last~Theorem. It~was~proposed}$$
$${\small \rm by~Leonhard~Euler~in}$$ $${\small 1769.}$$
$${\small \rm It~states~that~for~all~integers}$$ $${\small n}$$ $${\small \rm and}$$ $${\small k}$$
$${\small \rm greater~than}$$ $${\small 1}$$$${\small \rm ,~if~the~sum~of}$$ $${\small n}$$ $${\small \rm many}$$
$${\small k}$$th $${\small \rm powers~of~positive~integers~is~itself}$$
$${\small \rm a}$$ $${\small k}$$$${\small \rm th~power,~then}$$$${\small n}$$ $${\small \rm is~greater~than~or}$$
$${\small \rm equal~to}$$ $${\small k:}$$
$${\small a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}=b^{k}\implies n\geq k}$$
$${\small \rm The~conjecture~represents~an~attempt~to}$$
$${\small \rm generalize~Fermat's~Last~Theorem,}$$
$${\small \rm which~is~the~special~case}$$ $${\small n = 2:}$$
$${\small \rm if}$$ $${\small a_{1}^{k}+a_{2}^{k}=b^{k},}$$$${\small \rm then}$$ $${\small 2 ≥ k.}$$
$${\small \rm Although~the~conjecture~holds for}$$
$${\small \rm the~case}$$ $${\small k =3}$$$${\small \rm (which~follows~from~Fermat's}$$
$${\small \rm Last~Theorem~for~the~third~powers),}$$
$${\small \rm it~was~disproved~for}$$ $${\small k = 4}$$ $${\small \rm and}$$ $${\small k = 5.}$$
$${\small \rm It~is~unknown~whether~the~conjecture~fails}$$
$${\small \rm or~holds~for~any~value}$$ $${\small k ≥ 6.}$$
$${\small k = 4}$$
$${\small \rm In}$$ $${\small 1988,}$$ $${\small \rm Noam~Elkies~published~a~method}$$
$${\small \rm to~construct~an~infinite~sequence~of}$$
$${\small \rm counterexamples~for~the}$$ $${\small k = 4}$$ $${\small \rm case.}$$
$${\small \rm His~smallest~counterexample~was}$$
$${\footnotesize 20615673^{4}=2682440^{4}+15365639^{4}+18796760^{4}.}$$
$${\small \rm In}$$$${\small 1988, }$$$${\small \rm Roger~Frye~found~the~smallest}$$
$${\small \rm possible~counterexample}$$
$${\footnotesize 95800^{4}+217519^{4}+414560^{4}=422481^{4}}$$
$${\small \rm for}$$ $${\small k = 4}$$ $${\small \rm by~a~direct~computer~search}$$
$${\small \rm using~techniques~suggested~by~Elkies.}$$
$${\small \rm This~solution~is~the~only~one~with~values~of}$$
$${\small \rm the~variables~below~1,000,000.}$$
$${\small k = 5}$$
$${\small \rm Euler's~conjecture~was~disproven~by~L. J. Lander}$$
$${\small \rm and~T. R. Parkin~in}$$ $${\small 1966 }$$$${\small \rm when,~through~a~direct}$$
$${\small \rm computer~search~on~a~CDC}$$ $${\small 6600,}$$ $${\small \rm they~found}$$
$${\small \rm a~counterexample~for}$$$${\small k = 5.}$$
$${\small \rm This~was~published~in~a~paper~comprising}$$
$${\small \rm just~two~sentences.}$$
$${\small \rm A~total~of~three~primitive~(that~is, in~which}$$
$${\small \rm the~summands~do~not~all~have~a~common}$$
$${\small \rm factor)~counterexamples~are~known:}$$
$${\small \displaystyle {\begin{aligned}144^{5}&=27^{5}+84^{5}+110^{5}+133^{5}\\14132^{5}&=(-220)^{5}+5027^{5}+6237^{5}+14068^{5}\\85359^{5}&=55^{5}+3183^{5}+28969^{5}+85282^{5}\end{aligned}}}$$
$${\small \rm (Lander~and~Parkin,~1966)}$$
$${\small \rm (Scher~and~Seidl,~1996)}$$
$${\small \rm (Frye,~2004).}$$
下記サイトに何がどこまでわかっているかが載っていますので参照して下さい😊
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Limits of search range
(検索範囲)
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\small \rm The~notation~is:}$$
$${\small (k, m, n)}$$
$${\small \rm where:}$$
$${\small k = }$$$${\small \rm power}$$
$${\small m = }$$$${\small \rm number~of~left~terms}$$
$${\small n = }$$$${\small \rm number~of~right~terms}$$
$${\small \rm *x~means~that~we~have~to~add}$$$${\smallx}$$
$${\small \rm times~the~number.}$$
$${\footnotesize (k,1,n)→n≦6000,b≦b_k~,~\rm Complete}$$
$${\footnotesize (k,b_k)}$$
$${\footnotesize (11,80),(12,52),(13,19),(14,43),(15,37)\\(16,28),(17,30),(18,26),(19,24),(20,20)}$$
$${\footnotesize (k,1,k+1)→b≦b_k~,~\rm Complete}$$
$${\footnotesize (k,b_k)}$$
$${\footnotesize (11,222)~\rm No~solution}$$
$${\footnotesize (14,174)~\rm No~solution}$$
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Search time required
(検索所要時間)
$${\footnotesize (11,1,23)~72~:~68202~\rm seconds}$$
$${\footnotesize (11,1,21)~75~:~48706~\rm seconds}$$
$${\footnotesize (11,1,21)~80~:~47404~\rm seconds}$$
$${\footnotesize (11,1,21)~76~:~45027~\rm seconds}$$
$${\footnotesize (13,1,233)~19~:~485207~\rm~seconds}$$
$${\footnotesize (13,1,248)~18~:~111665~\rm~seconds}$$
$${\footnotesize (13,1,262)~17~:~47611~\rm seconds}$$
$${\footnotesize (13,1,279)~16~:~11699~\rm seconds}$$
$${\footnotesize (14,1,39)~43~:~17812~\rm seconds}$$
$${\footnotesize (16,1,80)~28~:~10451~\rm seconds}$$
$${\footnotesize (17,1,79)~29~:~15428~\rm seconds}$$
$${\footnotesize (11,1,12)~222~:~18286~\rm seconds}$$
$${\footnotesize (14,1,15)~174~:~96468~\rm seconds}$$
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Current search range
(現在の検索中の範囲)
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\footnotesize (11,1,n)→n≦6000,b=81〜90~(4/24/24〜)}$$★
$${\footnotesize (12,1,n)→n≦6000,b=53〜60~(4/○/24〜)}$$
$${\footnotesize (13,1,n)→n≦6000,b=20〜30~(4/○/24〜)}$$
$${\footnotesize (14,1,n)→n≦6000,b=44〜50~(4/○/24〜)}$$
$${\footnotesize (15,1,n)→n≦6000,b=37〜40~(4/○/24〜)}$$
$${\footnotesize (16,1,n)→n≦6000,b=29〜30~(4/○/24〜)}$$
$${\footnotesize (17,1,n)→n≦6000,b=31〜40~(4/○/24〜)}$$
$${\footnotesize (18,1,n)→n≦6000,b=27〜30~(4/○/24〜)}$$
$${\footnotesize (19,1,n)→n≦6000,b=24〜30~(4/○/24〜)}$$
$${\footnotesize (20,1,n)→n≦6000,b=21〜30~(4/○/24〜)}$$
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Record Database
(記録のデータベース)
$${\footnotesize (11,1,15)}$$
$${\footnotesize 249=237+215+210+182+173+170+135+125+ 68*2+42*2+39+32+29 }$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 11/29/2003)}$$
$${\footnotesize(12,1,19)}$$
$${\footnotesize 1051=942+936+932+910+806+758+646+636+588+572+504+436+426+250+208+120+62+42+27}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 02/21/2005)}$$
$${\footnotesize(13,1,21)}$$
$${\footnotesize 274=259+247+235+210*2+208+205+180+175+162+157+150+149+106+100+80+58+45+26*2+16}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 04/20/2003)}$$
$${\footnotesize(14,1,25)}$$
$${\footnotesize 112=100+98*3+96+94+89+88*2+80*2+71+70*2+62+53+49+48*2+47+37+33+27+14+6}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 09/23/2005)}$$
$${\footnotesize(15,1,28)}$$
$${\footnotesize 197=185+178+170+169+166+165+155+135+127+125+122+120*2+105+99+95*2+87+76+75+68+64+41+38+36+22+19+10}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 09/05/2003)}$$
$${\footnotesize(16,1,49)}$$
$${\footnotesize 2601=2598+1972+1870+1564*2+1530+1496+1495+1462+1394+1350+1292+1258+1172+1138+1122*2+1056+1038+1008+884*3+868+818+816*2+748+714*3+650+646*2+636+522+408*2+340+238*2+210+204+170+134+68+52+34+32}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 04/21/2004)}$$
$${\footnotesize (17,1,39)}$$
$${\footnotesize 208=196+189*2+182+176+171+167+165+138+133+129+126*3+113+111+105+101+98*2+91+87+84+83+81*2+77+70+48+42+40+39+33+28+21+14+10+9+1}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 05/26/2004)}$$
$${\footnotesize(18,1,57)}$$
$${\footnotesize 76=73*2+60*2+54+50+45+44+41*4+40*4+33*6+27*3+25+24*3+18*2+17+13*2+12+11*2+9*2+8+6*6+5*6+3*2+2*3}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 09/27/2002)}$$
$${\footnotesize(19,1,51)}$$
$${\footnotesize 71=68+65+64+63+61+59+57*2+56+48*4+46+45*2+42*3+40+39+37*2+36+35*3+33+31+30*3+23+20+18*2+15*2+12+11+10*2+9*3+8+7*2+6+4*2}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 02/06/2003)}$$
$${\footnotesize (20,1,61)}$$
$${\footnotesize 106=98*4+91+89*3+84*2+81+77*2+72+71+70*4+68+67+63*2+56*4+53*2+49*2+48+44+42*2+40+39+38+37*2+35+33*2+30+28*2+25*3+23+21+19+18+14*2+13+9+3*2+2+1}$$
$${\footnotesize \rm (Jaroslaw~Wroblewski, 08/09/2004)}$$
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Database (11≦k≦20)
$${\small b^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\small 11≦k≦20.}$$
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ここから先は
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