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

Answer 1

Serenity Johnson

No

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 #F_0#, it begins:

#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 #m, n >= 1#

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)#

#square#

Hence:

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

So: #F_(2n)# is divisible by #F_n# for any #n >= 1#.

If #n = 2# then #F_2 = 1# so that does not imply that #F_4# is composite - it is actually prime. But for any larger values of #n#, #F_(2n)# is composite.

So for any two consecutive Fibonacci numbers after #(F_4, F_5) = (3, 5)#, since one of them will have even index greater than #4#, it will be divisible by an earlier non-unit Fibonacci number greater than #F_2#.

e.g. #F_22 = 17711# is divisible by #F_11 = 89#

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Answer 2

Anna Allen

No, there are no other pairs of consecutive Fibonacci numbers where both numbers are prime, apart from 2 and 3, and 3 and 5. This is because as the Fibonacci sequence progresses, the numbers become increasingly larger, making it less likely for consecutive numbers to both be prime. It is a well-known conjecture that there are infinitely many prime numbers in the Fibonacci sequence, but specific pairs of consecutive primes are rare and difficult to find as the sequence progresses.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.