Each of X and Y is a positive integer, such that:

• X and Y are relatively prime, and:
• Each of (X² – 5)/Y and (y² – 5)/X is a positive integer.

Does there exist an infinite numbers of pairs (X,Y) satisfying the given conditions?
Give reasons for your answer.

..or, more precisely, Lucas Aid, my memory having been
refreshed by your number sequence, Charlie.

1, 4, 11, 29, 76, …. , are alternate terms of the Lucas sequence {L_i}:
which looks like this:  1, 3, 4, 7, 11, 18, 29, 47, 76, ….
(i.e. Fibonacci-like but with a different starting pair.)

We know that L_n = L_n-1 + L_n-2 , from which it can be deduced that

L_n = 3L_n-2 – L_n-4 which gives the rule for Charlie’s sequence, {C_i}

as:        C_n = 3C_n-1 – C_n-2             (1)        with  C₁ = 1 and C₂ = 4

Noting that C_n-1² – 5 = C_nC_n-2 when n = 3 (i.e. 4² – 5 = 1*11), and

also that            C_n² - 5 = C_nC_n-2 + C_n² – C_n-1²

                                    = C_n(3C_n-1 – C_n) + C_n² – C_n-1²   using (1)

                                    = C_n-1(3C_n – C_n-1)

                                    = C_n+1C_n-1

we have proved by induction that   C_n-1² – 5 = C_nC_n-2  is true for all

values of n >= 3. Since all the C values are integers (from (1)), it

follows that (C_n-1² – 5)/C_n and (C_n² – 5)/C_n-1 are always integers,

so every pair of consecutive C values satisfy the required condition.

Also, since C₁ and C₂ are coprime, it follows from (1) that C₂ and C₃

are coprime and, by induction, that all consecutive Cs are coprime.

Posted by Harry on 2015-12-31 17:52:16