\(96\mathbf{\color{blue}{\;=\;}}31+32+33\) (som van opeenvolgende gehele getallen)

\(96\mathbf{\color{blue}{\;=\;}}30+32+34\) (som van opeenvolgende pare getallen)

\(96\mathbf{\color{blue}{\;=\;}}5+7+9+11+13+15+17+19\mathbf{\color{blue}{\;=\;}}11+13+15+17+19+21\mathbf{\color{blue}{\;=\;}}21+23+25+27\mathbf{\color{blue}{\;=\;}}47+49\)

\(\qquad\;\,\)(som van opeenvolgende onpare getallen)

\(96\mathbf{\color{blue}{\;=\;}}\)het kleinste getal dat op vier verschillende wijzen kan worden geschreven als het verschil van twee kwadraten :

\(\qquad\;\,10^2-2^2\mathbf{\color{blue}{\;=\;}}11^2-5^2\mathbf{\color{blue}{\;=\;}}14^2-10^2\mathbf{\color{blue}{\;=\;}}25^2-23^2\)

\(96\mathbf{\color{blue}{\;=\;}}2^5+2^5+2^5\)

\(96=5!-4!\)

\(96=(0;4;4;8)\lower2pt{\Large{\color{teal}{➋}}}\to\{\#1\}\)

\(96\mathbf{\color{blue}{\;=\;}}2^3+2^3+2^3+2^3+4^3\mathbf{\color{blue}{\;=\;}}1^3+1^3+1^3+1^3+1^3+3^3+4^3\mathbf{\color{blue}{\;=\;}}1^3+1^3+2^3+2^3+2^3+2^3+2^3+3^3+3^3\mathbf{\color{blue}{\;=\;}}\)

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

\(96\mathbf{\color{blue}{\;=\;}}2^5+[2^6][4^3][8^2]\mathbf{\color{blue}{\;=\;}}2^7-2^5\mathbf{\color{blue}{\;=\;}}[5^4][25^2]-23^2\mathbf{\color{blue}{\;=\;}}10^2-2^2\mathbf{\color{blue}{\;=\;}}11^2-5^2\mathbf{\color{blue}{\;=\;}}14^2-10^2\)

\(96\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad\;\,5\) oplossingen bekend

\(\qquad\;\,\)References Sum of Three Cubes

\(\qquad\;\,\bbox[3px,border:1px solid]{14^3+20^3+(-22)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{10853^3+13139^3+(-15250)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{(-18038812)^3+(-11450026)^3+19461410^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{(-3745798253038)^3+2466821330939^3+3348538539149^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{(-27829788093331)^3+(-9203823199573)^3+28161376592534^3}\mathbf{\color{blue}{\;=\;}}\)

\(96\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad\;\,\bbox[3px,border:1px blue solid]{2^5+2^5+2^5+k^5+(-k)^5}\mathbf{\color{blue}{\;=\;}}\)(als niet van deze vorm dan \(z\gt200)\)

\(96^6=1*2*3*4*6*8*12*16*24*32*48*96~~\) (product van alle delers – zie bij 12.14 voor meer info)

Som der reciproken van partitiegetallen van \(96\) is \(1\) op \(96\) (zesennegentig) wijzen !!

Vier partities hebben unieke termen.

\(~~(1)~~\bbox[navajowhite,3px,border:1px solid]{96=2+5+7+10+30+42}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{5}}+{\Large\frac{1}{7}}+{\Large\frac{1}{10}}+{\Large\frac{1}{30}}+{\Large\frac{1}{42}}\)

\(~~(5)~~\bbox[navajowhite,3px,border:1px solid]{96=2+6+9+12+18+21+28}~~\) en \(~~1={\Large\frac{1}{2}}+{\Large\frac{1}{6}}+{\Large\frac{1}{9}}+{\Large\frac{1}{12}}+{\Large\frac{1}{18}}+{\Large\frac{1}{21}}+{\Large\frac{1}{28}}\)

\((17)~~\bbox[navajowhite,3px,border:1px solid]{96=3+4+5+14+15+20+35}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{4}}+{\Large\frac{1}{5}}+{\Large\frac{1}{14}}+{\Large\frac{1}{15}}+{\Large\frac{1}{20}}+{\Large\frac{1}{35}}\)

\((52)~~\bbox[navajowhite,3px,border:1px solid]{96=4+5+6+7+12+14+20+28}~~\) en \(~~1={\Large\frac{1}{4}}+{\Large\frac{1}{5}}+{\Large\frac{1}{6}}+{\Large\frac{1}{7}}+{\Large\frac{1}{12}}+{\Large\frac{1}{14}}+{\Large\frac{1}{20}}+{\Large\frac{1}{28}}\)

(OEIS A125726)

Als som met de vier operatoren \(+-*\;/\)

\(96=(18+3)+(18-3)+(18*3)+(18/3)\)

\(96=(24+1)+(24-1)+(24*1)+(24/1)\)

$$ 2~odd~primes \left[ \begin{matrix} &7&+&89\\ &13&+&83\\ &17&+&79\\ &23&+&73\\ &29&+&67\\ &37&+&59\\ &43&+&53 \end{matrix} \right. $$

$$ 3~primes \left[ \begin{matrix} &&\mathbf{2}&+&\mathbf{5}&+&\mathbf{89}\\ &&\mathbf{2}&+&\mathbf{11}&+&\mathbf{83}\\ &&\mathbf{2}&+&\mathbf{23}&+&\mathbf{71}\\ &&\mathbf{2}&+&\mathbf{41}&+&\mathbf{53}\\ &2&+&47&+&47 \end{matrix} \right. $$

\begin{align} 96^2&=\mathbf{9}21\mathbf{6}\\ 996^2&=\mathbf{99}201\mathbf{6}\\ 9996^2&=\mathbf{999}2001\mathbf{6}\\ 99996^2&=\mathbf{9999}20001\mathbf{6}\\ 999996^2&=\mathbf{99999}200001\mathbf{6}\\ 9999996^2&=\mathbf{999999}2000001\mathbf{6}\\ \cdots&=\cdots \end{align}

\(96^2\mathbf{\color{blue}{\;=\;}}[2^{10}][4^5][32^2]+2^{13}\mathbf{\color{blue}{\;=\;}}[10^4][100^2]-28^2\mathbf{\color{blue}{\;=\;}}104^2-40^2\mathbf{\color{blue}{\;=\;}}120^2-72^2\mathbf{\color{blue}{\;=\;}}146^2-110^2\mathbf{\color{blue}{\;=\;}}160^2-[2^{14}][4^7][128^2]\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~204^2-180^2\mathbf{\color{blue}{\;=\;}}265^2-247^2\mathbf{\color{blue}{\;=\;}}296^2-280^2\mathbf{\color{blue}{\;=\;}}390^2-378^2\mathbf{\color{blue}{\;=\;}}580^2-572^2\mathbf{\color{blue}{\;=\;}}771^2-765^2\mathbf{\color{blue}{\;=\;}}1154^2-1150^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~2305^2-2303^2\)

\(96^3\mathbf{\color{blue}{\;=\;}}24^5-192^3\mathbf{\color{blue}{\;=\;}}160^3-1792^2\mathbf{\color{blue}{\;=\;}}944^2-80^2\mathbf{\color{blue}{\;=\;}}960^2-192^2\mathbf{\color{blue}{\;=\;}}1056^2-480^2\mathbf{\color{blue}{\;=\;}}1120^2-608^2\mathbf{\color{blue}{\;=\;}}1240^2-808^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~1344^2-960^2\mathbf{\color{blue}{\;=\;}}1680^2-1392^2\mathbf{\color{blue}{\;=\;}}1856^2-[40^4][1600^2]\mathbf{\color{blue}{\;=\;}}2156^2-1940^2\mathbf{\color{blue}{\;=\;}}2400^2-2208^2\mathbf{\color{blue}{\;=\;}}3144^2-3000^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~3520^2-3392^2\mathbf{\color{blue}{\;=\;}}4150^2-4042^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{4656^2-4560^2}\mathbf{\color{blue}{\;=\;}}6180^2-6108^2\mathbf{\color{blue}{\;=\;}}6944^2-6880^2\mathbf{\color{blue}{\;=\;}}8219^2-8165^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~9240^2-9192^2\)

\(96={\Large\frac{16\;*\;17\;*\;18}{16~+~17~+~18}}~~\) (OEIS A001082) (OEIS A032766)

\(96\mathbf{\color{blue}{\;=\;}}\begin{align}\left({\frac{38}{39}}\right)^3+\left({\frac{178}{39}}\right)^3\mathbf{\color{blue}{\;=\;}}\color{tomato}{\left({\frac{19*2}{39}}\right)^3+\left({\frac{89*2}{39}}\right)^3\mathbf{\color{blue}{\;=\;}}\left({\frac{19}{39}}\right)^3+\left({\frac{89}{39}}\right)^3\mathbf{\color{blue}{\;=\;}}{\frac{96}{8}}\mathbf{\color{blue}{\;=\;}}12}\end{align}\)

(Integral Sum of Two Rational Cubes) (OEIS A020898) (OEIS A228499)

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

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

\(96\) is som van vijf kwadraten op vijf wijzen

\(96=1^2+1^2+2^2+3^2+9^2\)

\(96=1^2+1^2+3^2+6^2+7^2\)

\(96=1^2+3^2+5^2+5^2+6^2\)

\(96=2^2+2^2+4^2+6^2+6^2\)

\(96=2^2+3^2+3^2+5^2+7^2\)

\(96\) is de oppervlakte van twee driehoeken met gehele getallen als zijden \((12;16;20),(8;26;30)\)

(Formule van Heron)

 ○–○–○ 

Som Der Cijfers (\(sdc\)) van \(k^{\large{96}}\) is gelijk aan het grondtal \(k\). De triviale oplossingen \(0\) en \(1\) negerend vinden we :

\(\qquad\qquad~sdc\left(1387^{\large{96}}\right)=1387~~\to\) Unieke oplossing voor \(k\gt1\)

Expressie met tweemaal de cijfers uit het getal \(96\)
\(96=6*(prime(6)+9/\sqrt9)\)

Als expressie met enkelcijferige toepassing, resp. van \(1\) tot \(9~~\) (met dank aan Inder. J. Taneja).
\(\qquad\qquad96=(11+1)*(1+1)^{(1+1+1)}\)
\(\qquad\qquad96=2*2*(22+2)\)
\(\qquad\qquad96=3*33-3\)
\(\qquad\qquad96=4*(4*4+4+4)\)
\(\qquad\qquad96=5*(5*5-5)-5+5/5\)
\(\qquad\qquad96=66+6*6-6\)
\(\qquad\qquad96=7*(7+7)-(7+7)/7\)
\(\qquad\qquad96=88+8\)
\(\qquad\qquad96=99-(9+9+9)/9\)

Met de cijfers van \(1\) tot \(9\) in stijgende en dalende volgorde (met dank aan Inder. J. Taneja) :
\(\qquad\qquad96=1*2+3+4*5+6+7*8+9\)
\(\qquad\qquad96=9+8*7+6+5*4+3+2*1\)

\(\Huge\bbox[border:0]{⏮}\)

\(\Huge\bbox[border:0]{⯬}\)

\(\Huge\bbox[border:0]{⏴}\)

\(\Huge\bbox[border:0]{⏵}\)

\(\Huge\bbox[border:0]{⯮}\)

\(\Huge\bbox[border:0]{⏭}\)