Arithmetical study of recurrence sequences

Arithmetical Study of a Certain Ternary Recurrence Sequence and Related Questions

“…In the present paper we will often apply Theorem 1 of §11 of [6]. We describe its use in our situation: Let p be a prime / 2, 11 (these are the primes dividing the discriminant of the minimal polynomial of of).…”

“…The problem of the explicit computation of these solutions is a difficult one, and in a previous paper of ours [6] we propose a general practical method for the explicit solution of equations as above. The purpose of our present paper is to give an interesting application of our method [6] to the equation (1) bn = ±2r3°, where ibn)nez is Berstel's ternary recurrence sequence defined by b0 = bx=0, b2= I, bn+3 = 2bn+2 -4bn+x + 4bn.…”

“…We quote from the introduction of our paper [6]: "Among ternary linear recurrence sequences, it seems that Berstel's sequence...plays a very special role. Firstly, it is the only known example of a nondegenerate ternary linear recurrence sequence which has six zeros (by definition, a nondegenerate linear recurrence sequence has only finitely many zeros).…”

“…For the problem studied here, i.e., the equation u" = ±2T , it seems again that Berstel's sequence has remarkable properties: we can prove that there are exactly 44 solutions («, r, s)." In our aforementioned paper we announce without proof the complete solution of ( 1 ) (see the theorem in §IV of [6]). Here we will give all the details of the solution.…”

“…Here we will give all the details of the solution. In particular, we hope to make clear, by means of the concrete example which we study, the part of our method described only in general terms in the remark of §111 of [6].…”