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contructing positive integers when (a + b + 1) divides (a^2 + b^2 + 1) ?

Source: Czech-Polish-Slovak Junior Match 2014, Team p5 CPSJ

March 23, 2020
gamenumber theorydividesdivisor

Problem Statement

There is the number 11 on the board at the beginning. If the number aa is written on the board, then we can also write a natural number bb such that a+b+1a + b + 1 is a divisor of a2+b2+1a^2 + b^2 + 1. Can any positive integer appear on the board after a certain time? Justify your answer.
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