Show that there exist an infinite number of infinite sequences of distinct positive integers a, b, c, d, ... for which a+1, ab+1, abc+1, abcd+1, ... are all squares.
a= n^2-1 => a+1 = n^2
b= n^2+1=> ab+1 =n^4-1+1= n^4= (n^2)^2
c= n^4+1=> abc+1 = n^8+1-1= n^8= (n^4)^2
d= n^8+1=> abcd+1= n^16-1+1 = n^16 = (n^8)^2
Continuing like this, there exist an infinite number of infinite sequences of distinct positive integers a,b,c,d,.... for which each of a+1, an+1, abc+1, abcd+1,.... is a perfect square.
Puzzle solution
