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infinitely pairs of consecutive lucky numbers in number theory

Source: Mexican Mathematical Olympiad 1998 OMM P1

July 28, 2018
number theoryDigitsconsecutive

Problem Statement

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number 11. For example, 19001900 is lucky, as 1900→82→68→100→11900 \to 82 \to 68 \to 100 \to 1. Find infinitely many pairs of consecutive numbers each of which is lucky.
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