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a_n =3(n^2 + n) + 7 , NT

Source: Polish MO second round 1959 p4

August 29, 2024
number theoryperfect cubedivides

Problem Statement

Given a sequence of numbers 13,25,43,… 13, 25, 43, \ldots whose n n -th term is defined by the formula an=3(n2+n)+7a_n =3(n^2 + n) + 7 Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by 5 5 ,
2( No term of the sequence is the cube of an integer.
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