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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 a>b\sqrt3-1.

(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


If some pair (a,b) of positive integers satisfies (1) then a and b have the same parity,

ab+1 = ab+\frac{3b^2-a^2}2 = (a+b)\frac{3b-a}2, so a+b|ab+1,

ab-1 = ab-\frac{3b^2-a^2}2 = (a-b)\frac{a+3b}2, so a-b|ab-1;


\sqrt3 b = \sqrt{a^2+2} < \sqrt{a^2+2a+1} = a+1, so a>\sqrt3b-1.

The Pell-like equation (1) has infinitely many solutions which can be represented as

a_n\pm b_n\sqrt3 = \big(1\pm \sqrt3\big)\cdot\big(2\pm \sqrt3\big)^n,

a_n = \dfrac{
\big(1-\sqrt3\big)\cdot\big(2-\sqrt3\big)^n}2, & \cr
b_n = \dfrac{
& (n=0,1,2,\ldots).}

(The same pairs can be generated by the recurrence a0=b0=1, an+1=2an+3bn, bn+1=an+2bn.)

For n\ge1 we have bn>1.


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