¿Qué factor numérico tiene [tex]15 ^ {n + 2} + 16 ^ {2n + 1} [ / tex] , n∈N?

1) Veamos para n = 1

<img src="https://tex.z-dn.net/?f=f%28n%29%3D15%5E%7Bn%2B2%7D%20%20%2B%2016%5E%7B2n%2B1%7D%20%5C%5C%20%5C%5C%0Af%281%29%3D31%5Ccdot%20241" />

2) Intentemos "adivinar" cuáles de los dos factores primos son factores de<img src="https://tex.z-dn.net/?f=f%28n%29" />

2.

1)

<img src="https://tex.z-dn.net/?f=16%5E%7B2n%2B1%7D%5Cequiv%20%28-15%29%5E%7B2n%2B1%7D%5Cmod%2031%5C%5C%20%5C%5C%0A16%5E%7B2n%2B1%7D%5Cequiv%20-15%5E%7B2n%2B1%7D%5Cmod%2031%5C%5C%20%5C%5C%0A%5Ctext%7BLuego%20...%7D%5C%5C%20%5C%5C%0A15%5E%7Bn%2B2%7D%20%2B%2016%5E%7B2n%2B1%7D%5Cequiv%2015%5E%7Bn%2B2%7D%20-%2015%5E%7B2n%2B1%7D%5Cmod%2031%5C%5C%5C%5C%0A15%5E%7Bn%2B2%7D%20%2B%2016%5E%7B2n%2B1%7D%5Cequiv%2015%5E%7Bn%2B1%7D%2815%20-%2015%5E%7Bn%7D%29%5Cmod%2031%5C%5C%5C%5C%0A%5Ctexttt%7BPara%20n%3D1%2C%20tenemos%20residuo%200%2C%20pero%20para%20n%5C%20%5Ctextgreater%20%5C%201%2C%20no.%7D%5C%5C%0A%5Ctexttt%7BPor%20ende%20descartamos%20a%2031%20como%20factor%20de%20%7Df%28n%29%0A%0A" />

2.

2) Ahora vamos a suponer que<img src="https://tex.z-dn.net/?f=241%20%7C%20f%28n%29%5C%2C%2C%5C%2C%20%5Cforall%20n%5Cin%20%5Cmathbb%7BN%7D" />.

Veamos si es cierto<img src="https://tex.z-dn.net/?f=241%7Cf%28n%2B1%29" /> :

<img src="https://tex.z-dn.net/?f=f%28n%2B1%29%3D15%5E%7Bn%2B3%7D%2B16%5E%7B2n%2B3%7D%5C%5C%20%5C%5C%0A16%5E2%3D256%5Cequiv%2015%5Cmod%20241%5C%5C%20%5C%5C%0A16%5E%7B2n%2B1%7D%3D16%5E%7B2n%2B1%7D%5Cmod%20241%5C%5C%20%5C%5C%0A%5Ctexttt%7Bentonces%3A%20%7D16%5E2%5Ccdot16%5E%7B2n%2B1%7D%5Cequiv15%5Ccdot%2016%5E%7B2n%2B1%7D%5Cmod%20241%5C%5C%20%5C%5C%20%0A%5Ctexttt%7BPor%20consiguiente%3A%20%7D%5C%5C%20%5C%5C%0A15%5E%7Bn%2B3%7D%2B16%5E%7B2n%2B3%7D%5Cequiv%2015%5E%7Bn%2B3%7D%2B15%5Ccdot%2016%5E%7B2n%2B1%7D%5Cmod%20241%5C%5C%20%5C%5C%20%0A15%5E%7Bn%2B3%7D%2B16%5E%7B2n%2B3%7D%5Cequiv%2015%5Cleft%2815%5E%7Bn%2B2%7D%2B16%5E%7B2n%2B1%7D%5Cright%29%5Cmod%20241%5C%5C%20%5C%5C%20%0A15%5E%7Bn%2B3%7D%2B16%5E%7B2n%2B3%7D%5Cequiv%20%5Cleft%2815%5E%7Bn%2B2%7D%2B16%5E%7B2n%2B1%7D%5Cright%29%5Cmod%20241%5C%5C%20%5C%5C%20%0A%5Ctexttt%7BPor%20hip%5C%27otesis%20inductiva%3A%20%7D%0A15%5E%7Bn%2B3%7D%2B16%5E%7B2n%2B3%7D%5Cequiv0%5Cmod%20241%5C%5C%20%5C%5C%0Af%28n%2B1%29%5Cequiv0%5Cmod%20241" />

Por fin el factor buscado es 241.