Il Passatempo Fratto

[Pasquale]

Si, confermo: la somma delle frazioni deve essere un intero, quale che sia. La questione del 2n voleva indicare la quantità dei numeri in gioco, logicamente pari, metà dei quali utilizzati come numeratori e metà come denominatori. Normalmente, sono poco sintetico, magari ingenerando qualche qui pro quo.
Tuttavia, guardando anche al passato di Base5, penso sempre che potrebbero esservi dei giovani visitatori, che magari danno uno sguardo senza manifestarsi, per i quali qualche parola in più potrebbe risultare utile (spero).
Purtroppo, l''equivoco è stato indotto dall'esempio riportato, che fortuitamente era un 6, corrispondente proprio a 2n.

[Gianfranco]

...c'è ancora margine

Fenomenale, complimenti Sergio!

P.S.
Propongo di riportare i prossimi risultati anche in formato testo, oltre che LaTeX, perché così si possono riutilizzare più facilmente.

[Quelo]

P.S.
Propongo di riportare i prossimi risultati anche in formato testo, oltre che LaTeX, perché così si possono riutilizzare più facilmente.

Fatto

[Bruno]

Affinando la tecnica arriviamo a n = 56, ma c'è ancora margine

(...)

EDIT:

Infatti anche n = 76

(...)

Raccolgo anche la sfida di Panurgo con n = 55

(...)

Tutto tutto con Excel, Sergio?
Mirabile.

[Quelo]

Sì, fatto tutto con Excel e un piccolo supporto di Wolfram Alpha
Ma non è tanto laborioso, ci vuole solo un po' di pazienza

Prendo un numero che promette bene, con pochi fattori primi e molti divisori tipo $\displaystyle 55440 = 2^4*3^2*5*7*11$
Prendo tutti i divisori di 55440 da 2 a 990, sono 88 e li metto sulla riga 2 del foglio elettronico
Pendo 88 numeri primi, diciamo da 31 a 521 e li metto sulla riga 1
Sulla riga 3 la divisione e sulla 4 il prodotto della divisione per 55440
Sommo tutti i termini della riga 3 e ottengo 270,758
Sommo tutti i termini della riga 4 e ottengo 15010844

L'obiettivo è ottenere 15024240 = 271 * 55440, quindi devo colmalre la differenza di 13396 aumentando i numeratori
Ogni unità numeratore incrementa la somma di 55440/d
Poiché i denominatori sono quasi tutti pari, i numeratori devono rimanere dispari, quindi l'incremento minimo è di 2, che corrisponde a 110880/d, li metto nella quinta riga e sotto il rapporto tra la differenza 13396 e questi "passi" minimi
Vedo subito che non ci sono valori interi quindi vanno modificati almeno 2 numeratori
Qui bisogna andare un po' per tentativi, se modifico la 521/990 in 523/990 (irriducicbile) e la 457/616 in 459/616 (anche questa iriducibile) mi accorgo che la differenza rimasta pari a 13216 è multipla di 224, l'incrementatore della frazione 433/495
Aggingendo 118 a 433 ottengo 551/495 (irriducibile) ed il gioco è fatto: 88 termini

$\displaystyle \frac{31}{2}+\frac{37}{3}+\frac{41}{4}+\frac{43}{5}+\frac{47}{6}+\frac{53}{7}+\frac{59}{8}+\frac{61}{9}+\frac{67}{10}+\frac{71}{11}+\frac{73}{12}+\frac{79}{14}+\frac{83}{15}+\frac{89}{16}+\frac{97}{18}+\frac{101}{20}+\frac{103}{21}+\frac{107}{22}+\frac{109}{24}+\frac{113}{28}+\frac{127}{30}+\frac{131}{33}+\frac{137}{35}+\frac{139}{36}+ \\
\displaystyle \frac{149}{40}+\frac{151}{42}+\frac{157}{44}+\frac{163}{45}+\frac{167}{48}+\frac{173}{55}+\frac{179}{56}+\frac{181}{60}+\frac{191}{63}+\frac{193}{66}+\frac{197}{70}+\frac{199}{72}+\frac{211}{77}+\frac{223}{80}+\frac{227}{84}+\frac{229}{88}+\frac{233}{90}+\frac{239}{99}+\frac{241}{105}+\frac{251}{110}+\frac{257}{112}+\frac{263}{120}+ \\
\displaystyle \frac{269}{126}+\frac{271}{132}+\frac{277}{140}+\frac{281}{144}+\frac{283}{154}+\frac{293}{165}+\frac{307}{168}+\frac{311}{176}+\frac{313}{180}+\frac{317}{198}+\frac{331}{210}+\frac{337}{220}+\frac{347}{231}+\frac{349}{240}+\frac{353}{252}+\frac{359}{264}+\frac{367}{280}+\frac{373}{308}+\frac{379}{315}+\frac{383}{330}+\frac{389}{336}+\frac{397}{360}+ \\
\displaystyle \frac{401}{385}+\frac{409}{396}+\frac{419}{420}+\frac{421}{440}+\frac{431}{462}+\frac{551}{495}+\frac{439}{504}+\frac{443}{528}+\frac{449}{560}+\frac{459}{616}+\frac{461}{630}+\frac{463}{660}+\frac{467}{693}+\frac{479}{720}+\frac{487}{770}+\frac{491}{792}+\frac{499}{840}+\frac{503}{880}+\frac{509}{924}+\frac{523}{990}=271$

Codice: Seleziona tutto

31/2+37/3+41/4+43/5+47/6+53/7+59/8+61/9+67/10+71/11+73/12+79/14+83/15+89/16+97/18+101/20+103/21+107/22+109/24+113/28+127/30+131/33+137/35+139/36+149/40+151/42+157/44+163/45+167/48+173/55+179/56+181/60+191/63+193/66+197/70+199/72+211/77+223/80+227/84+229/88+233/90+239/99+241/105+251/110+257/112+263/120+269/126+271/132+277/140+281/144+283/154+293/165+307/168+311/176+313/180+317/198+331/210+337/220+347/231+349/240+353/252+359/264+367/280+373/308+379/315+383/330+389/336+397/360+401/385+409/396+419/420+421/440+431/462+551/495+439/504+443/528+449/560+459/616+461/630+463/660+467/693+479/720+487/770+491/792+499/840+503/880+509/924+523/990=271

[Quelo]

Mi ero ripromesso di arrivare a 100 termini e con ulteriori elaborazioni è fattibile

Prendiamo $\displaystyle 2^3*3^3*5^2*7*11=415800$
I divisori da 2 a 990 sono 108, altrettanti numeri primi (es. da 31 a 647) saranno i numeratori
La somma delle frazioni è 311,7386, il numeratore della frazione non ridotta è 129662077
Per arrivare all'intero più vicino 312 mancano 67523 unità
Come abbiamo visto i numeratori si incrementano a passi di 2, quindi ogni contributo è un numero pari, per cui la differenza non po’ essere colmata.
Il problema si risolve facilmente eliminando una frazione che fornisce un contributo dispari, per esempio quella con il denominatore 840

Ora la somma delle frazioni è 311,1 e la somma al numeratore è 129355672
Per arrivare a 311 bisogna sottrarre $\displaystyle 41872 = 2^4*2617$
Per farlo dobbiamo trovare 2 o 3 "incrementatori" il cui MCD sia divisore di 41872
Potremmo scegliere 1200 e 1232 ma non si perviene ad una soluzione, prendiamo invece 1200, 1232 e 1386 che corrispondono alle frazioni 593/693, 587/675 e 563/600
L'equazione $1200x+1232y+1386z=-41872$ ha soluzioni intere parametriche, scelgo quella che mi permette di modulare le frazioni mantenendole irriducibili: $x=-116, y=52, z=24$
Le frazioni interessate diventano pertanto 361/693, 691/675 e 611/600
Abbiamo così una soluzione per $n = 107$, quale sarà il prossimo traguardo, 125?

$\displaystyle \frac{31}{2}+\frac{37}{3}+\frac{41}{4}+\frac{43}{5}+\frac{47}{6}+\frac{53}{7}+\frac{59}{8}+\frac{61}{9}+\frac{67}{10}+\frac{71}{11}+\frac{73}{12}+\frac{79}{14}+\frac{83}{15}+\frac{89}{18}+\frac{97}{20}+\frac{101}{21}+\frac{103}{22}+\frac{107}{24}+\frac{109}{25}+\frac{113}{27}+\frac{127}{28}+\frac{131}{30}+\frac{137}{33}+\frac{139}{35}+ \\
\displaystyle \frac{149}{36}+\frac{151}{40}+\frac{157}{42}+\frac{163}{44}+\frac{167}{45}+\frac{173}{50}+\frac{179}{54}+\frac{181}{55}+\frac{191}{56}+\frac{193}{60}+\frac{197}{63}+\frac{199}{66}+\frac{211}{70}+\frac{223}{72}+\frac{227}{75}+\frac{229}{77}+\frac{233}{84}+\frac{239}{88}+\frac{241}{90}+\frac{251}{99}+\frac{257}{100}+\frac{263}{105}+ \\
\displaystyle \frac{269}{108}+\frac{271}{110}+\frac{277}{120}+\frac{281}{126}+\frac{283}{132}+\frac{293}{135}+\frac{307}{140}+\frac{311}{150}+\frac{313}{154}+\frac{317}{165}+\frac{331}{168}+\frac{337}{175}+\frac{347}{180}+\frac{349}{189}+\frac{353}{198}+\frac{359}{200}+\frac{367}{210}+\frac{373}{216}+\frac{379}{220}+\frac{383}{225}+\frac{389}{231}+ \\
\displaystyle \frac{397}{252}+\frac{401}{264}+\frac{409}{270}+\frac{419}{275}+\frac{421}{280}+\frac{431}{297}+\frac{433}{300}+\frac{439}{308}+\frac{443}{315}+\frac{449}{330}+\frac{457}{350}+\frac{461}{360}+\frac{463}{378}+\frac{467}{385}+\frac{479}{396}+\frac{487}{420}+\frac{491}{440}+\frac{499}{450}+\frac{503}{462}+\frac{509}{495}+\frac{521}{504}+ \\
\displaystyle \frac{523}{525}+\frac{541}{540}+\frac{547}{550}+\frac{557}{594}+\frac{611}{600}+\frac{569}{616}+\frac{571}{630}+\frac{577}{660}+\frac{691}{675}+\frac{361}{693}+\frac{599}{700}+\frac{601}{756}+\frac{607}{770}+\frac{613}{792}+\frac{617}{825}+\frac{631}{900}+\frac{641}{924}+\frac{643}{945}+\frac{647}{990}=311$

Codice: Seleziona tutto

31/2+37/3+41/4+43/5+47/6+53/7+59/8+61/9+67/10+71/11+73/12+79/14+83/15+89/18+97/20+101/21+103/22+107/24+109/25+113/27+127/28+131/30+137/33+139/35+149/36+151/40+157/42+163/44+167/45+173/50+179/54+181/55+191/56+193/60+197/63+199/66+211/70+223/72+227/75+229/77+233/84+239/88+241/90+251/99+257/100+263/105+269/108+271/110+277/120+281/126+283/132+293/135+307/140+311/150+313/154+317/165+331/168+337/175+347/180+349/189+353/198+359/200+367/210+373/216+379/220+383/225+389/231+397/252+401/264+409/270+419/275+421/280+431/297+433/300+439/308+443/315+449/330+457/350+461/360+463/378+467/385+479/396+487/420+491/440+499/450+503/462+509/495+521/504+523/525+541/540+547/550+557/594+611/600+569/616+571/630+577/660+691/675+361/693+599/700+601/756+607/770+613/792+617/825+631/900+641/924+643/945+647/990=311

... e intanto $119$ li abbiamo fatti

$\displaystyle \frac{31}{2}+\frac{37}{3}+\frac{41}{4}+\frac{43}{5}+\frac{47}{6}+\frac{53}{7}+\frac{59}{8}+\frac{61}{9}+\frac{67}{10}+\frac{71}{12}+\frac{73}{14}+\frac{79}{15}+\frac{83}{16}+\frac{89}{18}+\frac{97}{20}+\frac{101}{21}+\frac{103}{24}+\frac{107}{25}+\frac{109}{27}+\frac{113}{28}+\frac{127}{30}+\frac{131}{32}+\frac{137}{35}+\frac{139}{36}+\frac{149}{40}+\frac{151}{42}+\frac{157}{45}+\frac{163}{48}+\frac{167}{49}+\frac{173}{50}+\frac{179}{54}+\frac{181}{56}+\frac{191}{60}+ \\ \displaystyle \frac{193}{63}+\frac{197}{70}+\frac{199}{72}+\frac{211}{75}+\frac{223}{80}+\frac{227}{81}+\frac{229}{84}+\frac{233}{90}+\frac{239}{96}+\frac{241}{98}+\frac{251}{100}+\frac{257}{105}+\frac{263}{108}+\frac{269}{112}+\frac{271}{120}+\frac{277}{125}+\frac{281}{126}+\frac{283}{135}+\frac{293}{140}+\frac{307}{144}+\frac{311}{147}+\frac{313}{150}+\frac{317}{160}+\frac{331}{162}+\frac{337}{168}+\frac{347}{175}+\frac{349}{180}+\frac{353}{189}+\frac{359}{196}+\frac{367}{200}+ \\ \displaystyle \frac{373}{210}+\frac{379}{216}+\frac{383}{224}+\frac{389}{225}+\frac{397}{240}+\frac{401}{245}+\frac{409}{250}+\frac{419}{252}+\frac{421}{270}+\frac{431}{280}+\frac{433}{288}+\frac{439}{294}+\frac{443}{300}+\frac{449}{315}+\frac{457}{324}+\frac{461}{336}+\frac{463}{350}+\frac{467}{360}+\frac{479}{375}+\frac{487}{378}+\frac{491}{392}+\frac{499}{400}+\frac{503}{405}+\frac{509}{420}+\frac{521}{432}+\frac{523}{441}+\frac{541}{450}+\frac{547}{480}+\frac{557}{490}+ \\ \displaystyle \frac{563}{500}+\frac{569}{504}+\frac{571}{525}+\frac{577}{540}+\frac{587}{560}+\frac{593}{567}+\frac{599}{588}+\frac{601}{600}+\frac{607}{630}+\frac{613}{648}+\frac{617}{672}+\frac{619}{675}+\frac{631}{700}+\frac{641}{720}+\frac{643}{735}+\frac{647}{750}+\frac{653}{756}+\frac{659}{784}+\frac{999}{800}+\frac{731}{810}+\frac{677}{840}+\frac{691}{875}+\frac{775}{882}+\frac{709}{900}+\frac{719}{945}+\frac{727}{980}+\frac{733}{1000}=319$

Codice: Seleziona tutto

31/2+37/3+41/4+43/5+47/6+53/7+59/8+61/9+67/10+71/12+73/14+79/15+83/16+89/18+97/20+101/21+103/24+107/25+109/27+113/28+127/30+131/32+137/35+139/36+149/40+151/42+157/45+163/48+167/49+173/50+179/54+181/56+191/60+193/63+197/70+199/72+211/75+223/80+227/81+229/84+233/90+239/96+241/98+251/100+257/105+263/108+269/112+271/120+277/125+281/126+283/135+293/140+307/144+311/147+313/150+317/160+331/162+337/168+347/175+349/180+353/189+359/196+367/200+373/210+379/216+383/224+389/225+397/240+401/245+409/250+419/252+421/270+431/280+433/288+439/294+443/300+449/315+457/324+461/336+463/350+467/360+479/375+487/378+491/392+499/400+503/405+509/420+521/432+523/441+541/450+547/480+557/490+563/500+569/504+571/525+577/540+587/560+593/567+599/588+601/600+607/630+613/648+617/672+619/675+631/700+641/720+643/735+647/750+653/756+659/784+999/800+731/810+677/840+691/875+775/882+709/900+719/945+727/980+733/1000=319

[panurgo]

$\displaystyle \frac23+\frac45+\frac8{15}=2$

[Quelo]

Mi piace l'approccio minimalista

$\displaystyle \frac{2}{5}+\frac{3}{7}+\frac{6}{35}=1$

Nel frattempo il record è salito a $164$

$\displaystyle \frac{31}{2}+\frac{37}{3}+\frac{61}{4}+\frac{67}{5}+\frac{73}{6}+\frac{97}{7}+\frac{101}{8}+\frac{113}{9}+\frac{127}{10}+\frac{137}{11}+\frac{151}{12}+\frac{157}{13}+\frac{163}{14}+\frac{167}{15}+\frac{173}{16}+\frac{17}{18}+\frac{19}{20}+\frac{191}{21}+\frac{193}{22}+\frac{23}{24}+\frac{199}{25}+\frac{211}{26}+\frac{223}{27}+\frac{227}{28}+\frac{29}{30}+\frac{32}{33}+\frac{34}{35}+\frac{229}{36}+\frac{38}{39}+\frac{241}{40}+\frac{41}{42}+\frac{43}{44}+\frac{257}{45}+\frac{47}{48}+ \\
\displaystyle \frac{49}{50}+\frac{51}{52}+\frac{53}{54}+\frac{271}{55}+\frac{277}{56}+\frac{59}{60}+\frac{62}{63}+\frac{64}{65}+\frac{281}{66}+\frac{69}{70}+\frac{71}{72}+\frac{74}{75}+\frac{76}{77}+\frac{283}{78}+\frac{79}{80}+\frac{83}{84}+\frac{87}{88}+\frac{89}{90}+\frac{293}{91}+\frac{98}{99}+\frac{313}{100}+\frac{103}{104}+\frac{317}{105}+\frac{107}{108}+\frac{109}{110}+\frac{111}{112}+\frac{116}{117}+\frac{119}{120}+\frac{125}{126}+\frac{129}{130}+\frac{131}{132}+\frac{134}{135}+ \\
\displaystyle \frac{139}{140}+\frac{142}{143}+\frac{331}{144}+\frac{149}{150}+\frac{153}{154}+\frac{155}{156}+\frac{164}{165}+\frac{167}{168}+\frac{174}{175}+\frac{337}{176}+\frac{179}{180}+\frac{181}{182}+\frac{188}{189}+\frac{194}{195}+\frac{197}{198}+\frac{199}{200}+\frac{207}{208}+\frac{209}{210}+\frac{215}{216}+\frac{219}{220}+\frac{224}{225}+\frac{230}{231}+\frac{233}{234}+\frac{239}{240}+\frac{251}{252}+\frac{259}{260}+\frac{263}{264}+\frac{269}{270}+\frac{272}{273}+ \\
\displaystyle \frac{274}{275}+\frac{279}{280}+\frac{285}{286}+\frac{296}{297}+\frac{299}{300}+\frac{307}{308}+\frac{311}{312}+\frac{314}{315}+\frac{324}{325}+\frac{329}{330}+\frac{335}{336}+\frac{349}{350}+\frac{347}{351}+\frac{359}{360}+\frac{363}{364}+\frac{377}{378}+\frac{384}{385}+\frac{389}{390}+\frac{395}{396}+\frac{399}{400}+\frac{419}{420}+\frac{428}{429}+\frac{431}{432}+\frac{439}{440}+\frac{449}{450}+\frac{454}{455}+\frac{461}{462}+\frac{467}{468}+\frac{494}{495}+ \\
\displaystyle \frac{503}{504}+\frac{519}{520}+\frac{524}{525}+\frac{527}{528}+\frac{539}{540}+\frac{545}{546}+\frac{549}{550}+\frac{559}{560}+\frac{571}{572}+\frac{584}{585}+\frac{593}{594}+\frac{599}{600}+\frac{615}{616}+\frac{623}{624}+\frac{629}{630}+\frac{649}{650}+\frac{659}{660}+\frac{674}{675}+\frac{692}{693}+\frac{699}{700}+\frac{701}{702}+\frac{714}{715}+\frac{719}{720}+\frac{727}{728}+\frac{787}{756}+\frac{769}{770}+\frac{779}{780}+\frac{833}{792}+\frac{818}{819}+ \\
\displaystyle \frac{824}{825}+\frac{839}{840}+\frac{857}{858}+\frac{879}{880}+\frac{899}{900}+\frac{909}{910}+\frac{923}{924}+\frac{935}{936}+\frac{944}{945}+\frac{956}{975}+\frac{991}{990}=421$

Codice: Seleziona tutto

31/2+37/3+61/4+67/5+73/6+97/7+101/8+113/9+127/10+137/11+151/12+157/13+163/14+167/15+173/16+17/18+19/20+191/21+193/22+23/24+199/25+211/26+223/27+227/28+29/30+32/33+34/35+229/36+38/39+241/40+41/42+43/44+257/45+47/48+49/50+51/52+53/54+271/55+277/56+59/60+62/63+64/65+281/66+69/70+71/72+74/75+76/77+283/78+79/80+83/84+87/88+89/90+293/91+98/99+313/100+103/104+317/105+107/108+109/110+111/112+116/117+119/120+125/126+129/130+131/132+134/135+139/140+142/143+331/144+149/150+153/154+155/156+164/165+167/168+174/175+337/176+179/180+181/182+188/189+194/195+197/198+199/200+207/208+209/210+215/216+219/220+224/225+230/231+233/234+239/240+251/252+259/260+263/264+269/270+272/273+274/275+279/280+285/286+296/297+299/300+307/308+311/312+314/315+324/325+329/330+335/336+349/350+347/351+359/360+363/364+377/378+384/385+389/390+395/396+399/400+419/420+428/429+431/432+439/440+449/450+454/455+461/462+467/468+494/495+503/504+519/520+524/525+527/528+539/540+545/546+549/550+559/560+571/572+584/585+593/594+599/600+615/616+623/624+629/630+649/650+659/660+674/675+692/693+699/700+701/702+714/715+719/720+727/728+787/756+769/770+779/780+833/792+818/819+824/825+839/840+857/858+879/880+899/900+909/910+923/924+935/936+944/945+956/975+991/990=421

... e niente, $204$

$\displaystyle \frac{61}{2}+\frac{97}{3}+\frac{113}{4}+\frac{127}{5}+\frac{151}{6}+\frac{163}{7}+\frac{173}{8}+\frac{191}{9}+\frac{193}{10}+\frac{211}{11}+\frac{223}{12}+\frac{227}{13}+\frac{229}{14}+\frac{241}{15}+\frac{257}{16}+\frac{277}{17}+\frac{19}{18}+\frac{281}{20}+\frac{283}{21}+\frac{23}{22}+\frac{293}{24}+\frac{313}{25}+\frac{317}{26}+\frac{331}{27}+\frac{29}{28}+\frac{31}{30}+\frac{32}{33}+\frac{337}{34}+\frac{347}{35}+\frac{37}{36}+\frac{38}{39}+\frac{41}{40}+\frac{43}{42}+ \\ \displaystyle \frac{353}{44}+\frac{46}{45}+\frac{47}{48}+\frac{49}{50}+\frac{367}{51}+\frac{53}{52}+\frac{379}{54}+\frac{383}{55}+\frac{57}{56}+\frac{59}{60}+\frac{62}{63}+\frac{64}{65}+\frac{67}{66}+\frac{69}{68}+\frac{71}{70}+\frac{73}{72}+\frac{74}{75}+\frac{76}{77}+\frac{79}{78}+\frac{81}{80}+\frac{83}{84}+\frac{86}{85}+\frac{87}{88}+\frac{89}{90}+\frac{92}{91}+\frac{98}{99}+\frac{101}{100}+\frac{103}{102}+\frac{397}{104}+\frac{106}{105}+\frac{107}{108}+\frac{109}{110}+ \\ \displaystyle \frac{111}{112}+\frac{116}{117}+\frac{118}{119}+\frac{121}{120}+\frac{125}{126}+\frac{129}{130}+\frac{131}{132}+\frac{134}{135}+\frac{137}{136}+\frac{139}{140}+\frac{142}{143}+\frac{145}{144}+\frac{149}{150}+\frac{152}{153}+\frac{155}{154}+\frac{157}{156}+\frac{164}{165}+\frac{167}{168}+\frac{169}{170}+\frac{174}{175}+\frac{177}{176}+\frac{179}{180}+\frac{181}{182}+\frac{186}{187}+\frac{188}{189}+\frac{194}{195}+\frac{197}{198}+\frac{199}{200}+\frac{203}{204}+ \\ \displaystyle \frac{207}{208}+\frac{209}{210}+\frac{215}{216}+\frac{219}{220}+\frac{222}{221}+\frac{224}{225}+\frac{230}{231}+\frac{233}{234}+\frac{237}{238}+\frac{239}{240}+\frac{251}{252}+\frac{254}{255}+\frac{259}{260}+\frac{263}{264}+\frac{269}{270}+\frac{271}{272}+\frac{274}{273}+\frac{276}{275}+\frac{279}{280}+\frac{285}{286}+\frac{296}{297}+\frac{299}{300}+\frac{305}{306}+\frac{307}{308}+\frac{311}{312}+\frac{314}{315}+\frac{324}{325}+\frac{329}{330}+\frac{335}{336}+ \\ \displaystyle \frac{339}{340}+\frac{349}{350}+\frac{352}{351}+\frac{356}{357}+\frac{359}{360}+\frac{363}{364}+\frac{373}{374}+\frac{377}{378}+\frac{384}{385}+\frac{389}{390}+\frac{395}{396}+\frac{399}{400}+\frac{407}{408}+\frac{419}{420}+\frac{424}{425}+\frac{428}{429}+\frac{431}{432}+\frac{439}{440}+\frac{441}{442}+\frac{449}{450}+\frac{454}{455}+\frac{458}{459}+\frac{461}{462}+\frac{467}{468}+\frac{475}{476}+\frac{494}{495}+\frac{503}{504}+\frac{509}{510}+\frac{519}{520}+ \\ \displaystyle \frac{524}{525}+\frac{527}{528}+\frac{539}{540}+\frac{545}{546}+\frac{549}{550}+\frac{559}{560}+\frac{562}{561}+\frac{571}{572}+\frac{584}{585}+\frac{593}{594}+\frac{596}{595}+\frac{599}{600}+\frac{611}{612}+\frac{615}{616}+\frac{623}{624}+\frac{629}{630}+\frac{649}{650}+\frac{659}{660}+\frac{662}{663}+\frac{674}{675}+\frac{679}{680}+\frac{692}{693}+\frac{699}{700}+\frac{701}{702}+\frac{713}{714}+\frac{716}{715}+\frac{719}{720}+\frac{727}{728}+\frac{747}{748}+ \\ \displaystyle \frac{755}{756}+\frac{764}{765}+\frac{769}{770}+\frac{401}{780}+\frac{409}{792}+\frac{421}{816}+\frac{433}{819}+\frac{443}{825}+\frac{457}{840}+\frac{463}{850}+\frac{479}{858}+\frac{489}{880}+\frac{491}{884}+\frac{499}{900}+\frac{521}{910}+\frac{523}{918}+\frac{541}{924}+\frac{547}{935}+\frac{557}{936}+\frac{563}{945}+\frac{775}{952}+\frac{601}{975}+\frac{587}{990}=643$

Codice: Seleziona tutto

61/2+97/3+113/4+127/5+151/6+163/7+173/8+191/9+193/10+211/11+223/12+227/13+229/14+241/15+257/16+277/17+19/18+281/20+283/21+23/22+293/24+313/25+317/26+331/27+29/28+31/30+32/33+337/34+347/35+37/36+38/39+41/40+43/42+353/44+46/45+47/48+49/50+367/51+53/52+379/54+383/55+57/56+59/60+62/63+64/65+67/66+69/68+71/70+73/72+74/75+76/77+79/78+81/80+83/84+86/85+87/88+89/90+92/91+98/99+101/100+103/102+397/104+106/105+107/108+109/110+111/112+116/117+118/119+121/120+125/126+129/130+131/132+134/135+137/136+139/140+142/143+145/144+149/150+152/153+155/154+157/156+164/165+167/168+169/170+174/175+177/176+179/180+181/182+186/187+188/189+194/195+197/198+199/200+203/204+207/208+209/210+215/216+219/220+222/221+224/225+230/231+233/234+237/238+239/240+251/252+254/255+259/260+263/264+269/270+271/272+274/273+276/275+279/280+285/286+296/297+299/300+305/306+307/308+311/312+314/315+324/325+329/330+335/336+339/340+349/350+352/351+356/357+359/360+363/364+373/374+377/378+384/385+389/390+395/396+399/400+407/408+419/420+424/425+428/429+431/432+439/440+441/442+449/450+454/455+458/459+461/462+467/468+475/476+494/495+503/504+509/510+519/520+524/525+527/528+539/540+545/546+549/550+559/560+562/561+571/572+584/585+593/594+596/595+599/600+611/612+615/616+623/624+629/630+649/650+659/660+662/663+674/675+679/680+692/693+699/700+701/702+713/714+716/715+719/720+727/728+747/748+755/756+764/765+769/770+401/780+409/792+421/816+433/819+443/825+457/840+463/850+479/858+489/880+491/884+499/900+521/910+523/918+541/924+547/935+557/936+563/945+775/952+601/975+587/990=643

SE&O

[panurgo]

$\displaystyle \frac23+\frac45+\frac8{15}=2$

Altro piccolo indizio:

$\displaystyle \frac67+\frac{10}{11}+\frac{18}{77}=2$

[panurgo]

Un altro indizio?

$\displaystyle \frac{12}{13}+\frac{16}{17}+\frac{30}{221}=2$

[Quelo]

$\displaystyle \frac{a-1}{a}+\frac{b-1}{b}+\frac{x}{ab}=2 \quad \Rightarrow \quad x=a+b$

o più in generale

$\displaystyle \frac{a-1}{a}+\frac{b-1}{b}+\frac{x}{ab}=c \quad \Rightarrow \quad x=ab(c-2)+a+b$

ma anche

$\displaystyle \frac{a-1}{a}+\frac{b-1}{b}+\frac{c-1}{c}+\frac{x}{abc}=d \quad \Rightarrow \quad x=abc(d-3)+ab+bc+ac$

e quindi

$\displaystyle \frac{a_1-1}{a_1}+\frac{a_2-1}{a_2}+\cdots+\frac{a_n-1}{a_n}+\frac{x}{\prod_{k=1}^n{a_k}}=b \quad \Rightarrow \quad x=\prod_{k=1}^n{a_k} \left [(b-n)+\sum_{j=1}^n \frac{1}{a_j} \right]$

[Bruno]

(...)
o più in generale

$\displaystyle \frac{a-1}{a}+\frac{b-1}{b}+\frac{x}{ab}=c \quad \Rightarrow \quad x=ab(c-2)+a+b$
(...)

Un'altra semplice generalizzazione:

$\displaystyle \frac{a-h}{a}+\frac{b-k}{b}+\frac{ab(h+k-2)+ak+bh}{ab} = h+k$

e quindi anche, fissando il 2 a destra:

$\displaystyle \frac{a-h}{a}+\frac{b-k}{b}+\frac{ak+bh}{ab} = 2.$

[Pasquale]

Chi l'avrebbe detto che quel pensierino venutomi in sogno avrebbe condotto a tanto ?
Al fin della vicenda, ognuno dei Grandi di Base5 avrà dato il suo contributo e maturato il diritto di appuntare al petto un numero primo, quale premio per il proprio impegno.
Intanto, se Quelo gradisce, avrei confezionato per lui un simpatico primo simmetrico: 1123211

[Bruno]

(...) un simpatico primo simmetrico: 1123211

Chissà se ne esiste qualcun altro dopo questo:

$\displaystyle \frac{1089\cdot 10^{101}+10^{205}-1}{9} = \underbrace{11 \cdot \cdot \cdot 11}_{101} \, 232 \, \underbrace{11 \cdot \cdot \cdot 11}_{101}$ .

[Pasquale]

Eh, questo è tuo, ma ti ci vuole una fascia da avvolgere intorno al torace