BULLETIN OF THE POLISH ACADEMY OF SCIENCES MATHEMATICS Vol. 64, No. 2-3, 2016 NUMBER THEORY On the Equation a 2 + bc = n with Restricted Unknowns by M. SKALBA Presented by Andrzej SCHINZEL To the memory of Professor Jerzy Browkin Summary. We extend our previous results concerning the equation a 2 + bc = n to all primes n and deal also with the general case of non-square n. Moreover, we provide partial results on patterns of ‘1’ and ‘11’ in the continued fractions of √ n. For a given positive integer n which is not a perfect square we are inter- ested in the triples of positive integers (a, b, c) satisfying the title equation (1) a 2 + bc = n and the restriction (2) b<c< √ n. The set of all such triples will be denoted by T (n) and their number by t(n). By trivial verification, t(3) = t(5) = t(7) = t(13) = t(23) = t(47) = 0 but T (11) = {(3, 1, 2)}, t(11) = 1. Similarly T (67) = {(5, 6, 7), (7, 3, 6), (8, 1, 3)}, t(67) = 3. The last two examples are emanations of a general phenomenon we have proved in [3]: if n is a prime of the form 8k +3 and n> 3 then t(n) is odd and positive a fortiori. 2010 Mathematics Subject Classification : 11E25, 11E16, 11A55. Key words and phrases : reduction of indefinite forms, continued fractions. Received 6 August 2016. Published online 14 December 2016. DOI: 10.4064/ba8079-11-2016 [137] c  Instytut Matematyczny PAN, 2016