Problem A. 551. (January 2012)
A. 551. Show that there exist infinitely many pairs (a,b) of positive integers with the property that a+b divides ab+1, a-b divides ab-1, b>1 and .
Deadline expired on 10 February 2012.
Solution (outline). In view of the solution of Problem A. 545., we look for positive integers a,b with the property
If some pair (a,b) of positive integers satisfies (1) then a and b have the same parity,
, so a+b|ab+1,
, so a-b|ab-1;
, so .
The Pell-like equation (1) has infinitely many solutions which can be represented as
(The same pairs can be generated by the recurrence a0=b0=1, an+1=2an+3bn, bn+1=an+2bn.)
For n1 we have bn>1.