\(75\mathbf{\color{blue}{\;=\;}}3+4+5+6+7+8+9+10+11+12\mathbf{\color{blue}{\;=\;}}10+11+12+13+14+15\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,13+14+15+16+17\mathbf{\color{blue}{\;=\;}}24+25+26\mathbf{\color{blue}{\;=\;}}37+38\) (som van opeenvolgende gehele getallen)

\(75\mathbf{\color{blue}{\;=\;}}11+13+15+17+19\mathbf{\color{blue}{\;=\;}}23+25+27\) (som van opeenvolgende onpare getallen)

\(75=3+5+7+11+13+17+19\) (som van opeenvolgende priemgetallen)

\(75=((0;1;5;7)\,(0;5;5;5)\,(1;1;3;8)\,(1;3;4;7)\,(3;4;5;5))\lower2pt{\Large{\color{teal}{➋}}}\to\{\#5\}\)

\(75\mathbf{\color{blue}{\;=\;}}1^3+1^3+1^3+2^3+4^3\mathbf{\color{blue}{\;=\;}}2^3+2^3+2^3+2^3+2^3+2^3+3^3\mathbf{\color{blue}{\;=\;}}1^3+1^3+1^3+1^3+1^3+2^3+2^3+3^3+3^3\mathbf{\color{blue}{\;=\;}}\)

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

\(75\mathbf{\color{blue}{\;=\;}}10^2-5^2\mathbf{\color{blue}{\;=\;}}14^2-11^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{38^2-37^2}\)

\(75\mathbf{\color{blue}{\;=\;}}\)(som van drie derdemachten)

\(\qquad\;\,4\) oplossingen bekend

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

\(\qquad\;\,\bbox[3px,border:1px solid]{4381159^3+435203083^3+(-435203231)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{1217343443218^3+2576191140760^3+(-2663786047493)^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{(-47258398396091)^3+(-47819328945509)^3+59897299698355^3}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad\;\,\bbox[3px,border:1px solid]{(-141757929834791)^3+(-186060602458637)^3+210217593792199^3}\mathbf{\color{blue}{\;=\;}}\)

\(75\mathbf{\color{blue}{\;=\;}}\)(som van vijf vijfdemachten)

\(\qquad\;\,\bbox[3px,border:1px blue solid]{20^5+(-42)^5+182^5+187^5+(-212)^5}\mathbf{\color{blue}{\;=\;}}(z\gt213)\)

\(\qquad\;\,\bbox[3px,border:1px blue solid]{39^5+101^5+183^5+267^5+(-275)^5}\)

\(\qquad\;\,\bbox[3px,border:1px blue solid]{(-41)^5+106^5+(-243)^5+(-330)^5+343^5}\)

$$ {\small{ 2\;primes \left[ \begin{matrix} \\ &2&+&73\\ \\ \end{matrix} \right. }} $$

$$ {\small{ 3\;primes \left[ \begin{matrix} &2&+&2&+&71\\ &&\mathbf{3}&+&\mathbf{5}&+&\mathbf{67}\\ &&\mathbf{3}&+&\mathbf{11}&+&\mathbf{61}\\ &&\mathbf{3}&+&\mathbf{13}&+&\mathbf{59}\\ &&\mathbf{3}&+&\mathbf{19}&+&\mathbf{53}\\ &&\mathbf{3}&+&\mathbf{29}&+&\mathbf{43}\\ &&\mathbf{3}&+&\mathbf{31}&+&\mathbf{41}\\ &&\mathbf{5}&+&\mathbf{11}&+&\mathbf{59}\\ &&\mathbf{5}&+&\mathbf{17}&+&\mathbf{53}\\ &&\mathbf{5}&+&\mathbf{23}&+&\mathbf{47}\\ &&\mathbf{5}&+&\mathbf{29}&+&\mathbf{41}\\ &7&+&7&+&61\\ &&\mathbf{7}&+&\mathbf{31}&+&\mathbf{37}\\ &11&+&11&+&53\\ &&\mathbf{11}&+&\mathbf{17}&+&\mathbf{47}\\ &&\mathbf{11}&+&\mathbf{23}&+&\mathbf{41}\\ &&\mathbf{13}&+&\mathbf{19}&+&\mathbf{43}\\ &13&+&31&+&31\\ &17&+&17&+&41\\ &17&+&29&+&29\\ &19&+&19&+&37\\ &23&+&23&+&29\\ \end{matrix} \right. }} $$

De som van de delers van \(75\), met uitsluiting van \(1\) en \(75\) zelf, is \(3+5+15+25=48\).
De som van de delers van \(48\), met uitsluiting van \(1\) en \(48\) zelf, is \(2+3+4+6+8+12+16+24=75\).
Alhoewel \(48\) en \(75\) dus niet officieel “bevriend” zijn, kan men zeggen dat ze toch affiniteit voor elkaar
hebben (sommige auteurs spreken zelfs van “verloofde” getallen! (Wikipedia : Betrothed numbers) (OEIS A005276)

Als som met de vier operatoren \(+-*\;/\)
\(75=(12+4)+(12-4)+(12*4)+(12/4)\)

\(75^2\mathbf{\color{blue}{\;=\;}}5^5+50^2\mathbf{\color{blue}{\;=\;}}[5^6][25^3][125^2]-[10^4][100^2]\mathbf{\color{blue}{\;=\;}}21^2+72^2\mathbf{\color{blue}{\;=\;}}45^2+60^2\mathbf{\color{blue}{\;=\;}}85^2-40^2\mathbf{\color{blue}{\;=\;}}195^2-180^2\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~317^2-308^2\mathbf{\color{blue}{\;=\;}}325^2-10^5\mathbf{\color{blue}{\;=\;}}565^2-560^2\mathbf{\color{blue}{\;=\;}}939^2-936^2\mathbf{\color{blue}{\;=\;}}2813^2-2812^2\)

\(75^3\mathbf{\color{blue}{\;=\;}}650^2-[5^4][25^2]\mathbf{\color{blue}{\;=\;}}750^2-375^2\mathbf{\color{blue}{\;=\;}}1050^2-825^2\mathbf{\color{blue}{\;=\;}}1630^2-1495^2\mathbf{\color{blue}{\;=\;}}1750^2-1625^2\mathbf{\color{blue}{\;=\;}}\bbox[2px,border:1px brown dashed]{2850^2-2775^2}\mathbf{\color{blue}{\;=\;}}\)

\(\qquad~~~4710^2-4665^2\mathbf{\color{blue}{\;=\;}}7826^2-7799^2\mathbf{\color{blue}{\;=\;}}8450^2-8425^2\)

\(75^2=53^2+54^2-10^2\)

De eerste keer dat er \(75\) opeenvolgende samengestelde getallen voorkomen gebeurt tussen de priemgetallen \(212701\)
en \(212777\) met aldus een priemkloof van \(76\,.~~\) (OEIS A000101.pdf)

\(2\)\(^{75}\)\(\,+\,75~~\) is een priemgetal, de zevende in zijn soort. (OEIS A052007)

\(\begin{align}75\mathbf{\color{blue}{\;=\;}}\left({\frac{17351}{3606}}\right)^3-\left({\frac{11951}{3606}}\right)^3\mathbf{\color{blue}{\;=\;}}\left({\frac{31401958790537999}{24991605897324612}}\right)^3+\left({\frac{104456145984145201}{24991605897324612}}\right)^3\end{align}\)

(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)

Som der reciproken van partitiegetallen van \(75\) is \(1\) op negenentwintig wijzen.

Eén partitie heeft unieke termen.

\(\bbox[navajowhite,3px,border:1px solid]{75=3+4+5+8+15+40}~~\) en \(~~1={\Large\frac{1}{3}}+{\Large\frac{1}{4}}+{\Large\frac{1}{5}}+{\Large\frac{1}{8}}+{\Large\frac{1}{15}}+{\Large\frac{1}{40}}\)

(OEIS A125726)