\(112=13+14+15+16+17+18+19\) (som van opeenvolgende gehele getallen)

\(112\mathbf{\color{blue}{\;=\;}}10+12+14+16+18+20+22\) (som van opeenvolgende pare getallen)

\(112\mathbf{\color{blue}{\;=\;}}7+9+11+13+15+17+19+21\mathbf{\color{blue}{\;=\;}}25+27+29+31\mathbf{\color{blue}{\;=\;}}55+57\)

\(\qquad~~~~\)(som van opeenvolgende onpare getallen)

\(112\mathbf{\color{blue}{\;=\;}}11+13+17+19+23+29\mathbf{\color{blue}{\;=\;}}53+59\) (som van opeenvolgende priemgetallen)

\(112=1*2+2*3+3*4+4*5+5*6+6*7\)

\(112=(6*7*8)/3\)

\(112=((2;2;2;10)\,(2;6;6;6)\,(4;4;4;8))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#3\}\)

\(112=((0;0;2;2;2;2;2;2;4)\,(0;1;1;1;1;3;3;3;3)\,(1;1;1;1;1;2;2;3;4))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#3\}\)

\(112=11^2-3^2\)

\(112=2^4+2^5+2^6\)

\(112\mathbf{\color{blue}{\;=\;}}2^4*7\)

\(112\mathbf{\color{blue}{\;=\;}}2^7-[2^4][4^2]\mathbf{\color{blue}{\;=\;}}[2^8][4^4][16^2]-12^2\mathbf{\color{blue}{\;=\;}}[2^9][8^3]-20^2\mathbf{\color{blue}{\;=\;}}2^{11}-44^2\mathbf{\color{blue}{\;=\;}}2^{19}-724^2\mathbf{\color{blue}{\;=\;}}11^2-3^2\mathbf{\color{blue}{\;=\;}}12^2-2^5\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~16^2-12^2\mathbf{\color{blue}{\;=\;}}29^2-[9^3][27^2]\)

\(112\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~\)Getallen van de vorm \(~9m+4~\) of \(~9m+5~\) kunnen nooit als som van drie derdemachten geschreven worden.

\(\qquad~~~~\)In dit geval is \(m=12~~(+4)\).

\(112\mathbf{\color{blue}{\;=\;}}\)(som van vier derdemachten)

\(\qquad~~~~(z\gt1000)\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-2)^3+(-2)^3+4^3+4^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{4^3+(-23)^3+(-26)^3+31^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{22^3+22^3+40^3+(-44)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{10^3+37^3+46^3+(-53)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{25^3+40^3+58^3+(-65)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-29)^3+(-44)^3+(-83)^3+88^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{34^3+(-53)^3+(-83)^3+88^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-2)^3+46^3+88^3+(-92)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{70^3+70^3+76^3+(-104)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{52^3+(-74)^3+(-101)^3+109^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-8)^3+52^3+106^3+(-110)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{4^3+(-86)^3+(-116)^3+130^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{25^3+115^3+121^3+(-149)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-101)^3+(-107)^3+(-137)^3+169^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-62)^3+118^3+163^3+(-179)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-23)^3+88^3+175^3+(-182)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{124^3+163^3+181^3+(-230)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{28^3+208^3+214^3+(-266)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-23)^3+(-83)^3+(-311)^3+313^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-146)^3+(-197)^3+(-275)^3+316^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{136^3+190^3+286^3+(-320)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-65)^3+205^3+301^3+(-329)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{94^3+184^3+316^3+(-338)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-14)^3+133^3+331^3+(-338)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{40^3+253^3+286^3+(-341)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{88^3+(-263)^3+(-305)^3+358^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{55^3+208^3+337^3+(-362)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-68)^3+265^3+328^3+(-377)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{139^3+274^3+373^3+(-422)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{121^3+124^3+415^3+(-422)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-104)^3+148^3+418^3+(-422)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{244^3+(-266)^3+(-419)^3+427^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{4^3+(-110)^3+(-470)^3+472^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-32)^3+190^3+472^3+(-482)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{295^3+298^3+409^3+(-494)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-26)^3+(-296)^3+(-458)^3+496^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{247^3+(-359)^3+(-464)^3+508^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{220^3+229^3+478^3+(-509)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-65)^3+184^3+529^3+(-536)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{52^3+97^3+592^3+(-593)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-116)^3+(-458)^3+(-488)^3+598^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-164)^3+229^3+598^3+(-605)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{115^3+184^3+604^3+(-611)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{178^3+(-416)^3+(-566)^3+628^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-194)^3+(-212)^3+(-626)^3+640^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-92)^3+211^3+637^3+(-644)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-2)^3+(-524)^3+(-530)^3+664^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{322^3+(-518)^3+(-572)^3+664^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{265^3+442^3+574^3+(-665)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-494)^3+(-509)^3+(-515)^3+730^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-218)^3+(-536)^3+(-650)^3+760^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-104)^3+(-305)^3+(-752)^3+769^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-338)^3+(-356)^3+(-722)^3+772^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{154^3+412^3+763^3+(-803)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-392)^3+658^3+673^3+(-809)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{118^3+(-602)^3+(-686)^3+814^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-290)^3+430^3+796^3+(-824)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{679^3+(-728)^3+(-815)^3+850^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-287)^3+(-617)^3+(-734)^3+868^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{130^3+517^3+802^3+(-869)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{1^3+(-296)^3+(-881)^3+892^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-221)^3+(-470)^3+(-878)^3+925^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{511^3+(-623)^3+(-884)^3+928^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{478^3+544^3+814^3+(-932)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-227)^3+706^3+790^3+(-941)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{406^3+(-608)^3+(-890)^3+952^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{241^3+241^3+961^3+(-971)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{22^3+(-608)^3+(-890)^3+976^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{97^3+(-320)^3+(-977)^3+988^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-356)^3+787^3+829^3+(-1004)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-449)^3+(-575)^3+(-962)^3+1054^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-104)^3+(-830)^3+(-932)^3+1114^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px green solid]{(-134)^3+(-818)^3+(-974)^3+1138^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~(z\gt1000)\)

\(112\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad~~~~\bbox[lightyellow,3px,border:1px blue solid]{~oplossing~onbekend~}\mathbf{\color{blue}{\;=\;}}\)

112.1
\(105^2+106^2+107^2+\cdots+112^2\mathbf{\color{blue}{\;=\;}}113^2+114^2+\cdots+119^2\)112.2

\(112*113=12656~~\) en \(~~211*311=65621\) (omgekeerden)

\(112^2=12544\) en het omgekeerde \(44521=211^2\). Hetzelfde doet zich voor bij

\(112^2\mathbf{\color{blue}{\;=\;}}[12^4][144^2]-2^{13}\mathbf{\color{blue}{\;=\;}}12^3+104^2\mathbf{\color{blue}{\;=\;}}20^4-384^2\mathbf{\color{blue}{\;=\;}}113^2-15^2\mathbf{\color{blue}{\;=\;}}130^2-66^2\mathbf{\color{blue}{\;=\;}}140^2-84^2\mathbf{\color{blue}{\;=\;}}212^2-180^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\,\,238^2-210^2\mathbf{\color{blue}{\;=\;}}400^2-384^2\mathbf{\color{blue}{\;=\;}}455^2-[21^4][441^2]\mathbf{\color{blue}{\;=\;}}788^2-780^2\)

\(112^3\mathbf{\color{blue}{\;=\;}}[2^{21}][8^7][128^3]-832^2\mathbf{\color{blue}{\;=\;}}28^3+1176^2\mathbf{\color{blue}{\;=\;}}224^3-[56^4][3136^2]\mathbf{\color{blue}{\;=\;}}448^3-9408^2\mathbf{\color{blue}{\;=\;}}1176^2+28^3\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\,\,1198^2-174^2\mathbf{\color{blue}{\;=\;}}1232^2-336^2\mathbf{\color{blue}{\;=\;}}1261^2-57^3\mathbf{\color{blue}{\;=\;}}1288^2-504^2\mathbf{\color{blue}{\;=\;}}1367^2-681^2\mathbf{\color{blue}{\;=\;}}1628^2-1116^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\,\,1792^2-1344^2\mathbf{\color{blue}{\;=\;}}1988^2-1596^2\mathbf{\color{blue}{\;=\;}}2872^2-2616^2\mathbf{\color{blue}{\;=\;}}3248^2-3024^2\mathbf{\color{blue}{\;=\;}}3682^2-3486^2\mathbf{\color{blue}{\;=\;}}5552^2-5424^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\,\,6272^2-336^3\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{6328^2-6216^2}\mathbf{\color{blue}{\;=\;}}7217^2-7119^2\)

112.3
  EEN KRINGLOOP  

\(112^2=012544~~\) en \(~~012+544=556~;~556^2=309136~~\) en \(~~309+136=445~;~445^2=198025~~\) en
\(198+025=223~;~223^2=049729~~\) en \(~~049+729=778~;~778^2=605284~~\) en \(~~605+284=889~;~889^2=790321\)
en \(~~790+321=001111~~\) en \(~~001+111=112~~\) terug bij het beginpunt. Een kortere kringloop heeft men met
\(334^2=111556~~\) en \(~~111+556=667~;~667^2=444889~~\) en \(~~444+889=001333\). Tenslotte is met
\(001+333=334~~\) de kring rond.

112.4
\(112\) als resultaat met breuken waarin de cijfers van \(0\) tot \(9\) exact één keer voorkomen : (\(2\) oplossingen) :
\(415296/3708=830592/7416=112\)
112.5
Men moet \(112\) tot minimaal de \(39691\)ste macht verheffen opdat in de decimale expansie exact \(112\) \(112\)'s verschijnen.
Terloops : \(112^{39691}\) heeft een lengte van \(81336\) cijfers.
112.6
Schakelaar
\(\mathbf[0\gets\to1000\mathbf]\)
Allemaal Getallen


\(112\)\(2^4*7\)\(10\)\(248\)
\(1,2,4,7,8,14,16,28,56,112\)
\(1110000_2\)\(160_8\)\(70_{16}\)
   

Uit de collectie 'Allemaal Getallen' van Ir. Jos Heynderickx
Bewerking & Layout door Patrick De Geest (email)
Laatste update 11 april 2024