Problem 11508
(American Mathematical Monthly, Vol.117, May 2010) Proposed by M. Bencze (Romania).
Prove that for all positive integersk there are infinitely many positive integers n such that kn + 1 and(k + 1)n + 1 are both perfect squares.
Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
Let us consider the Pell equation
u2− k(k + 1)v2= 1 which has infinite positive solutions {(ui, vi)} such that
u1= 2k + 1, v1= 2 and ui+ vipk(k + 1) =
u1+ v1pk(k + 1)i
for i ≥ 1.
By letting
ni = ((ui+ kvi)2− 1)/k = (2k + 1)vi2+ 2uivi we have a sequence of positive integers {ni} such that
kni+ 1 = (ui+ kvi)2 and (k + 1)ni+ 1 = (ui+ (k + 1)vi)2.
