# Apart from #2, 3# and #3, 5# is there any pair of consecutive Fibonacci numbers which are both prime?

##### 1 Answer

No

#### Explanation:

The Fibonacci sequence is defined by:

#F_0 = 0#

#F_1 = 1#

#F_n = F_(n-2) + F_(n-1)" "# for#n > 1#

Starting with

#0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...#

Prove by induction that:

#F_(m+n) = F_(m-1)F_n+F_mF_(n+1)#

for any

**Base cases**

#F_(m+color(blue)(1)) = F_(m-1) + F_m = F_(m-1)F_(color(blue)(1)) + F_m F_(color(blue)(1)+1)#

#F_(m+color(blue)(2)) = F_(m+1) + F_m = F_(m-1) + 2F_m = F_(m-1)F_color(blue)(2) + F_mF_(color(blue)(2)+1)#

**Induction step**

#F_(m+k+1) = F_(m+k-1) + F_(m+k)#

#color(white)(F_(m+k+1)) = F_(m-1)F_(k-1) + F_mF_k + F_(m-1)F_k + F_mF_(k+1)#

#color(white)(F_(m+k+1)) = F_(m-1)(F_(k-1) + F_k) + F_m(F_k +F_(k+1))#

#color(white)(F_(m+k+1)) = F_(m-1)F_(k+1) + F_m F_(k+2)#

Hence:

#F_(2n) = F_(n+n) = F_(n-1)F_n + F_nF_(n+1) = F_n(F_(n-1) + F_(n+1))#

So:

If

So for any two consecutive Fibonacci numbers after

e.g.