Question

How do you show that there are infinitely many triples `(a, b, c)` of integers such that `ab+1`, `bc+1` and `ca+1` are all perfect squares?

Answer 1

See explanation...

Explanation:

Define the sequence:

`a_0 = 0`

`a_1 = 1`

`a_(n+1) = 4a_n-a_(n-1)`

This is A001353 in the online encyclopedia of integer sequences.

The first few terms are:

`0, 1, 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120`

Then let:

`(a, b, c)_n = (a_n, 2a_(n+1), a_(n+2))` for `n = 1,2,3,...`

Then:

`a_n + a_(n+2) = a_n + (4a_(n+1)-a_n) = 4a_(n+1) = 2(2a_(n+1))`

So `a_n, 2a_(n+1), a_(n+2)` are in arithmetic progression.

We find:

`color(blue)(a_n b_n + 1 = (a_(n+1)-a_n)^2)`

`color(blue)(b_n c_n + 1 = (a_(n+2)-a_(n+1))^2)`

`color(blue)(c_n a_n + 1 = a_(n+1)^2)`

`color(white)()`
Proof by induction

`underline("Base case")`

`color(blue)(a_1 b_1 + 1) = 1*8 + 1 = 9 = (4-1)^2 = color(blue)((a_2 - a_1)^2)`

`underline("Induction steps")`

If:

`a_n b_n + 1 = (a_(n+1)-a_n)^2`

This expands to:

`2 a_n a_(n+1) + 1 = a_(n+1)^2-2 a_n a_(n+1) + a_n^2`

So:

`a_(n+1)^2 = 4a_n a_(n+1) - a_n^2 + 1`

`=(4 a_(n+1)-a_n)a_n + 1`

`=a_(n+2) * a_n + 1`

`=c_n a_n + 1`

So:

`color(blue)(a_n b_n + 1 = (a_(n+1)-a_n)^2 => c_n a_n + 1 = a_(n+1)^2)`

If:

`2 a_n a_(n+1) + 1 = (a_(n+1)-a_n)^2`

Then we find:

`(a_(n+2)-a_(n+1))^2`

`=(3a_(n+1)-a_n)^2`

`=9a_(n+1)^2-6a_n a_(n+1)+a_n^2`

`=(8a_(n+1)^2-2a_na_(n+1)+1)+(a_(n+1)^2-4a_na_(n+1)+a_n^2-1)`

`=(2a_(n+1)(4a_(n+1)-a_n)+1)+((a_(n+1)-a_n)^2-(2a_na_(n+1)+1))`

`=2a_(n+1)a_(n+2)+1`

`=b_n c_n+1`

That is:

`color(blue)(a_n b_n + 1 = (a_(n+1)-a_n)^2 => b_n c_n + 1 = (a_(n+2)-a_(n+1))^2)`

Then:

`color(blue)(a_(n+1) b_(n+1) + 1) = b_n c_n + 1 = 2 a_(n+1) a_(n+2) + 1 = color(blue)((a_(n+2)-a_(n+1))^2)`

`color(white)()`
Footnote

Iteratively defined sequences similar to the one for `a_n` above seem to be fairly common when finding sets of numbers with a particular property.

For example, the sequence of square roots of triangular square numbers is given by the recurrence relation:

`a_0 = 0`

`a_1 = 1`

`a_(n+1) = 6a_n - a_(n-1)`

The first few elements of this sequence are:

`0, 1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, 46611179, 271669860, 1583407981, 9228778026, 53789260175`

This is A001109 in the online encyclopedia of integer sequences.