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a_m and a_n relatively prime for m≠n

Source: Bulgaria 1973 P1

June 20, 2021

number theoryrelatively primeSequenceslimitsalgebra

Problem Statement

Let the sequence

a_1,a_2,\ldots,a_n,\ldots

is defined by the conditions:

a_1=2

and

a_{n+1}=a_n^2-a_n+1

(n=1,2,\ldots)

. Prove that:
(a)

a_m

and

a_n

are relatively prime numbers when

m\ne n

. (b)

\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1

I. Tonov