MathDB

Problem 1

Part of 1973 Bulgaria National Olympiad

Problems(1)

a_m and a_n relatively prime for m≠n

Source: Bulgaria 1973 P1

6/20/2021
Let the sequence a1,a2,…,an,…a_1,a_2,\ldots,a_n,\ldots is defined by the conditions: a1=2a_1=2 and an+1=an2−an+1a_{n+1}=a_n^2-a_n+1 (n=1,2,…)(n=1,2,\ldots). Prove that:
(a) ama_m and ana_n are relatively prime numbers when m≠nm\ne n. (b) lim⁡n→∞∑k=1n1ak=1\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1
I. Tonov
number theoryrelatively primeSequenceslimitsalgebra