Question

Calcula

\frac{\begin{gathered}(10^4+324)(22^4+324)\\{}\cdot(34^4+324)(46^4+324)\\{}\cdot(58^4+324)\end{gathered}}{\begin{gathered}(4^4+324)(16^4+324)\\{}\cdot(28^4+324)(40^4+324)\\{}\cdot(52^4+324)\end{gathered}}.

Compute

\frac{\begin{gathered}(10^4+324)(22^4+324)\\{}\cdot(34^4+324)(46^4+324)\\{}\cdot(58^4+324)\end{gathered}}{\begin{gathered}(4^4+324)(16^4+324)\\{}\cdot(28^4+324)(40^4+324)\\{}\cdot(52^4+324)\end{gathered}}.

Respuesta

Usa la identidad de Sophie Germain con

324=4\cdot3^4

Use Sophie Germain’s identity with

324=4\cdot3^4

Si

g(x)=x^2+6x+18,

reescribe

x^4+324

como

g(x-6)g(x)

If

g(x)=x^2+6x+18,

rewrite

x^4+324

as

g(x-6)g(x)

Sea

g(x)=x^2+6x+18.

La identidad de Sophie Germain da

x^4+324=g(x-6)g(x).

Por tanto, el numerador aporta

g(4),g(10),\ldots,g(58),

mientras que el denominador aporta

g(-2),g(4),\ldots,g(52).

Todo se cancela salvo

\begin{aligned}\frac{g(58)}{g(-2)}&=\frac{58^2+6(58)+18}{4-12+18}\\&=\frac{3730}{10}=373.\end{aligned}

Let

g(x)=x^2+6x+18.

Sophie Germain’s identity gives

x^4+324=g(x-6)g(x).

The numerator therefore supplies

g(4),g(10),\ldots,g(58),

while the denominator supplies

g(-2),g(4),\ldots,g(52).

Everything cancels except

\begin{aligned}\frac{g(58)}{g(-2)}&=\frac{58^2+6(58)+18}{4-12+18}\\&=\frac{3730}{10}=373.\end{aligned}