Advance P-23(BOMC-2,day 3)

[SANZEED]

Advance problem-23
Prove that for $n\ge 5$,$f_{n}+f_{n-1}-1$ has at least $n+1$ prime factors,where $f_{n}=2^{2^{n}}+1$.

[SANZEED]

My intuition said:

Try to factor the expression.

Hint:

Find a sequence regarding $f_{n}$

Now you may apply induction(How?)

[Tahmid Hasan]

These are fermat's numbers.So,hint.... and also the killer blow

$f_0.f_2........f_{n-1}+2=f_n$

[sourav das]

To help factorizing :

Sophie Germain Identity , or just factor $4a^4+b^4 $

Edited, Personal comment: Really really sorry . Thanks to *Mahi*

[Nadim Ul Abrar]

$(2^{2^n})^2+2.2^{2^n}+1-2^{2^n}=(2^{2^n}+1)^2-(2^{2^{n-1}})^2=(2^{2^n}+2^{2^{n-1}}+1)(2^{2^n}-2^{2^{n-1}}+1)$
induction

[*Mahi*]

Sophi Jermain

Please, when you write something about someone, please get their name right. If you don't know the correct spelling, just google it , and google will return you the correct spelling with "Did you mean "Sophie Germain Identity"?". They have worked hard for mathematics , and I'm sure they deserve this bit.