 Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# 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 .

(5 pont)

Deadline expired on February 10, 2012.

Solution (outline). In view of the solution of Problem A. 545., we look for positive integers a,b with the property

 a2-3b2=-2. (1)

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;

finally , 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 n 1 we have bn>1.

### Statistics:

 7 students sent a solution. 5 points: Ágoston Tamás, Gyarmati Máté, Janzer Olivér, Mester Márton, Omer Cerrahoglu, Strenner Péter, Szabó 789 Barnabás.

Problems in Mathematics of KöMaL, January 2012