\(264\mathbf{\color{blue}{\;=\;}}\)als som van opeenvolgende gehele getallen op drie verschillende wijzen :

\begin{cases} 264=9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24\\ 264=19+20+21+22+23+24+25+26+27+28+29\\ 264=87+88+89 \end{cases}

\(264\mathbf{\color{blue}{\;=\;}}\)als som van opeenvolgende pare getallen op drie verschillende wijzen :

\begin{cases} 264=14+16+18+20+22+24+26+28+30+32+34\\ 264=26+28+30+32+34+36+38+40\\ 264=86+88+90 \end{cases}

\(264\mathbf{\color{blue}{\;=\;}}\)als som van opeenvolgende onpare getallen op vier verschillende wijzen :

\begin{cases} 264=11+13+15+17+19+21+23+25+27+29+31+33\\ 264=39+41+43+45+47+49\\ 264=63+65+67+69\\ 264=131+133 \end{cases}

\(264\mathbf{\color{blue}{\;=\;}}11+13+17+19+23+29+31+37+41+43\) (som van opeenvolgende priemgetallen)

\(264\mathbf{\color{blue}{\;=\;}}((0;2;2;16)\,(0;2;8;14)\,(0;8;10;10)\,(2;4;10;12)\,(4;4;6;14)\,(6;8;8;10))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#6\}\)

\(264\mathbf{\color{blue}{\;=\;}}2^2+4^2+6^2+8^2+12^2\mathbf{\color{blue}{\;=\;}}3^2+5^2+7^2+9^2+10^2\)

\(264\mathbf{\color{blue}{\;=\;}}((0;0;0;0;2;4;4;4;4)\,(0;0;1;1;1;2;4;4;5)\,(0;0;2;2;2;2;2;2;6)\,(0;1;3;3;3;3;3;4;4)\)

\(\qquad~~~~(1;1;1;1;1;1;2;5;5)\,(1;1;1;1;1;2;2;3;6)\,(1;1;2;2;3;3;4;4;4)\,(2;2;2;2;2;2;3;4;5))\lower2pt{\Large{\color{teal}{➌}}}\to\{\#8\}\)

\(264\mathbf{\color{blue}{\;=\;}}2^1+6^1+4^4\mathbf{\color{blue}{\;=\;}}2^5+6^3+4^2\) (de grondtallen zijn de cijfers uit \(264\) cfrt. 373.1 ) (OEIS A050240)

\(264\mathbf{\color{blue}{\;=\;}}91+78/2+536/4\) (cijfers van \(1\) tot \(9\))

\(264\mathbf{\color{blue}{\;=\;}}26*(6+4)+4\mathbf{\color{blue}{\;=\;}}6^{6/2}+46+2\)

\(264\mathbf{\color{blue}{\;=\;}}eulerphi(299,335,483,536,598,644,670,804,828,966)\)

\(264\mathbf{\color{blue}{\;=\;}}2^3+[2^8][4^4][16^2]\mathbf{\color{blue}{\;=\;}}[5^4][25^2]-19^2\mathbf{\color{blue}{\;=\;}}17^2-5^2\mathbf{\color{blue}{\;=\;}}35^2-31^2\mathbf{\color{blue}{\;=\;}}67^2-65^2\)

\(264\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad~~~~4\) oplossingen bekend

\(\qquad~~~~\)References Sum of Three Cubes

\(\qquad~~~~\bbox[3px,border:1px solid]{430^3+(-1295)^3+1279^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{2254^3+2887^3+(-3287)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{17836^3+47977^3+(-48785)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\bbox[3px,border:1px solid]{(-75121098462971)^3+(-107059177620005)^3+118190407326250^3}\mathbf{\color{blue}{\;=\;}}\)

\(264\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad~~~~\)(fully searched up to \(z=1000)\)

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

\(264^2\mathbf{\color{blue}{\;=\;}}2^{13}+248^2\mathbf{\color{blue}{\;=\;}}265^2-23^2\mathbf{\color{blue}{\;=\;}}275^2-77^2\mathbf{\color{blue}{\;=\;}}286^2-110^2\mathbf{\color{blue}{\;=\;}}314^2-170^2\mathbf{\color{blue}{\;=\;}}330^2-198^2\mathbf{\color{blue}{\;=\;}}411^2-315^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;440^2-352^2\mathbf{\color{blue}{\;=\;}}520^2-448^2\mathbf{\color{blue}{\;=\;}}561^2-495^2\mathbf{\color{blue}{\;=\;}}750^2-702^2\mathbf{\color{blue}{\;=\;}}814^2-770^2\mathbf{\color{blue}{\;=\;}}986^2-950^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;1105^2-1073^2\mathbf{\color{blue}{\;=\;}}1464^2-1440^2\mathbf{\color{blue}{\;=\;}}1595^2-1573^2\mathbf{\color{blue}{\;=\;}}1945^2-1927^2\mathbf{\color{blue}{\;=\;}}2186^2-2170^2\mathbf{\color{blue}{\;=\;}}2910^2-2898^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;4360^2-4352^2\mathbf{\color{blue}{\;=\;}}5811^2-5805^2\mathbf{\color{blue}{\;=\;}}8714^2-8710^2\)

\(264^3\mathbf{\color{blue}{\;=\;}}4290^2-66^2\mathbf{\color{blue}{\;=\;}}4312^2-440^2\mathbf{\color{blue}{\;=\;}}4390^2-934^2\mathbf{\color{blue}{\;=\;}}4488^2-1320^2\mathbf{\color{blue}{\;=\;}}4620^2-1716^2\mathbf{\color{blue}{\;=\;}}4675^2-1859^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;4787^2-2125^2\mathbf{\color{blue}{\;=\;}}5060^2-2684^2\mathbf{\color{blue}{\;=\;}}5145^2-2841^2\mathbf{\color{blue}{\;=\;}}5313^2-3135^2\mathbf{\color{blue}{\;=\;}}5412^2-3300^2\mathbf{\color{blue}{\;=\;}}5720^2-3784^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;6188^2-4460^2\mathbf{\color{blue}{\;=\;}}6600^2-5016^2\mathbf{\color{blue}{\;=\;}}7062^2-5610^2\mathbf{\color{blue}{\;=\;}}7238^2-5830^2\mathbf{\color{blue}{\;=\;}}8338^2-7150^2\mathbf{\color{blue}{\;=\;}}8562^2-7410^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~~\;9240^2-8184^2\mathbf{\color{blue}{\;=\;}}9988^2-9020^2\mathbf{\color{blue}{\;=\;}}\cdots\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{34980^2-34716^2}\)

\(264^2\mathbf{\color{blue}{\;=\;}}48^2+144^2+216^2\mathbf{\color{blue}{\;=\;}}56^2-614^2+666^2\mathbf{\color{blue}{\;=\;}}909^2+924^2-1269^2\mathbf{\color{blue}{\;=\;}}\cdots\)

\(264\) is een getal dat gelijk is aan \(22\) maal de som van zijn cijfers : \(264=22*(2+6+4)\)

Andere getallen met dezelfde eigenschap zijn \(132\) en \(396~~\) (OEIS A005349 - Harshad getallen)

\((N^{10}-1)\) is deelbaar door \(264\) als \(N\) een priemgetal en groter dan \(3\) is.

Er zijn twee rechthoekige driehoeken met omtrek \(264\) en gehele zijden : \((22;120;122)\) en \((66;88;110)\)

Als som met de vier operatoren \(+-*\;/\)
\(264=(66+1)+(66-1)+(66*1)+(66/1)\)

\(\begin{align}264\mathbf{\color{blue}{\;=\;}}\left({\frac{1853}{291}}\right)^3+\left({\frac{523}{291}}\right)^3\end{align}\)

(Integral Sum of Two Rational Cubes) (OEIS A020898) (OEIS A228499) (Links uit OEIS A060838)

\((x^3+y^3)/z^3=n~\to~\) [x waarde] (OEIS A190356)  [y waarde] (OEIS A190580)  [z waarde] (OEIS A190581)

Kleinste positieve oplossingen \(~\to~\) [x waarde] (OEIS A254326)  [y waarde] (OEIS A254324)

\(b\mathbf{\color{blue}{\;=\;}}264\to\)
\(b\)\(^{1}\)\(+b\)\(^{8}\)\(+b\)\(^{7}\)\(+b\)\(^{5}\)\(+b\)\(^{8}\)\(+b\)\(^{6}\)\(+b\)\(^{0}\)\(+b\)\(^{9}\)\(+b\)\(^{2}\)\(+b\)\(^{2}\)\(+b\)\(^{6}\)\(+b\)\(^{4}\)\(+b\)\(^{2}\)\(+b\)\(^{2}\)\(+b\)\(^{1}\)\(+b\)\(^{1}\)\(+b\)\(^{2}\)\(+b\)\(^{3}\)\(+b\)\(^{9}\)\(+b\)\(^{6}\)\(+b\)\(^{8}\)\(+b\)\(^{9}\)\(+b\)\(^{0}\)\(\mathbf{\color{blue}{\;=\;}}\)
\(18758609226422112396890~~\)(OEIS A236067)

 ○–○–○ 

Som Der Cijfers (\(sdc\)) van \(k^{\large{264}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(2980^{\large{264}}\right)=2980\qquad\qquad~sdc\left(4275^{\large{264}}\right)=4275\qquad\qquad~sdc\left(4338^{\large{264}}\right)=4338\)

\(\qquad\qquad~sdc\left(4437^{\large{264}}\right)=4437\)

Expressie met tweemaal de cijfers uit het getal \(264\) enkel met operatoren \(+,-,*,/,(),\)^^\(\)
\(264\mathbf{\color{blue}{\;=\;}}(2\)^^\(6\)^^\(4)+2-6+4\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}(1+1)*11*(11+1)\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}2*22*(2+2+2)\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}33*(3*3-3/3)\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}4^4+4+4\mathbf{\color{blue}{\;=\;}}444-44*4-4\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}5*55-55/5\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}6*(6*6+6)+6+6\mathbf{\color{blue}{\;=\;}}666-66*6-6\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}7*7*7-77-(7+7)/7\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}(8+8)*(8+8)+8\)
\(\qquad\qquad264\mathbf{\color{blue}{\;=\;}}9*(9+9)-9+999/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad264=1+2+34+5*6*7+8+9\)
\(\qquad\qquad264=9+8+7*6*5+4+32+1\)

Alle getallen van \(1\) tot \(24\) komen aan bod in deze expressie van \(264\)
\(2+3+4+5+6+7+8+9+10+12+13+14+15+16+17+18+19+20+21+22+23\mathbf{\color{blue}{\;=\;}}1*11*24\)

\(264\) is een getal gelijk aan de som van permutaties, twee aan twee, van deelverzamelingen van zijn eigen cijfers.

\(\qquad264\mathbf{\color{blue}{\;=\;}}24+26+42+46+62+64\)

(Digit-reassembly numbers or Osiris numbers) 132.28 396.23

\(264\) is het grootst gekende getal wiens kwadraat palindromisch èn golvend is (Eng. undulating).
\(\qquad264^2\mathbf{\color{blue}{\;=\;}}69696\)

(vijf multigrades) \(264\to9006349824\to\)

\begin{aligned} 264^1&={\color{green}{8}}^1+{\color{tomato}{34}}^1+62^1+68^1+{\color{brown}{92}}^1\\ 264^5&={\color{green}{8}}^5+{\color{tomato}{34}}^5+62^5+68^5+{\color{brown}{92}}^5\\ \\ 264^1&={\color{green}{8}}^1+41^1+47^1+79^1+89^1\\ 264^5&={\color{green}{8}}^5+41^5+47^5+79^5+89^5\\ \\ 264^1&=12^1+18^1+72^1+78^1+84^1\\ 264^5&=12^5+18^5+72^5+78^5+84^5\\ \\ 264^1&=21^1+{\color{tomato}{34}}^1+43^1+74^1+{\color{brown}{92}}^1\\ 264^5&=21^5+{\color{tomato}{34}}^5+43^5+74^5+{\color{brown}{92}}^5\\ \\ 264^1&=24^1+42^1+48^1+54^1+96^1\\ 264^5&=24^5+42^5+48^5+54^5+96^5\\ \end{aligned}

(vijf multigrades) \(264\to264^5\to\)

\begin{aligned} 264^1&=52^1-72^1-176^1+204^1+256^1\\ 264^5&=52^5-72^5-176^5+204^5+256^5\\ \\ 264^1&=-136^1-461^1+903^1+1267^1-1309^1\\ 264^5&=-136^5-461^5+903^5+1267^5-1309^5\\ \\ 264^1&=-385^1-440^1+1199^1+1397^1-1507^1\\ 264^5&=-385^5-440^5+1199^5+1397^5-1507^5\\ \\ 264^1&=693^1+1419^1-2046^1-2244^1+2442^1\\ 264^5&=693^5+1419^5-2046^5-2244^5+2442^5\\ \\ 264^1&=-312^1-888^1+1512^1+2328^1-2376^1\\ 264^5&=-312^5-888^5+1512^5+2328^5-2376^5\\ \end{aligned}

Kleinste oplossing voor de positieve Pell vergelijking \(x^2-D*y^2\mathbf{\color{blue}{\;=\;}}1~\) met \(D\mathbf{\color{blue}{\;=\;}}264\).

Als \(D\) een kwadraat is dan zijn er geen oplossingen.

\({\qquad\color{darkviolet}{65}}^2-264*{\color{darkviolet}{4}}^2\mathbf{\color{blue}{\;=\;}}1\)

(Pell equation solver)

⏮

⯬

⏴

⏵

⯮

⏭