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?
1 Answer
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
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)#
Proof by induction
#color(blue)(a_1 b_1 + 1) = 1*8 + 1 = 9 = (4-1)^2 = color(blue)((a_2 - a_1)^2)#
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)#
Footnote
Iteratively defined sequences similar to the one for
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.
