For each N in N let Fn 22n 1 These are called the Fermat n

For each N in N, let F_n = 2^2^n + 1. These are called the Fermat numbers after the French mathematician Pierre de Fermat (1601-1665). Fermat showed that F_0, F_1, F_2, F_3 and F_4 are prime and conjectured that all Fermat numbers are prime. However, over a 100 years later, Euler showed that F_5 is not prime. It is not known if there is any n > 4 for which F_n is prime. Prove that for all n > 1, F_n = (F_0 * F_1 * F_2....F_n+1) + 2.

Solution

In example there is one mistak noot n+1 is n-1

to prove that for all n>=1

F_n=(F₀*F₁*------F_n-1)+2

we prove this by using mathematical induction

step 1) we prove for n=1

we have F₀+2= 3+2=5= F₁

this prove for n =1

induction hypothysis

we assume for n =k

F_k=(F₀*F₁*------F_k-1)+2

to prove feo n=k+1

consider

(F₀*F₁*------F_k)+2= (F₀*F₁*------F_k-1F_k)+2

                       = (F_k-2)F_k+2

                        = (2^{2^k}-1)(2^{2^k}+1)+2

                         =2^{2^k}+1

                           =F_k+1

this prove for n=k+1

by induction

all n>=1

F_n=(F₀*F₁*------F_n-1)+2 is porve