Le Monde puzzle [#1063] and le bac’

As pointed out by Jean-Louis Foulley in his comment on the recent Le Monde puzzle [#1063] post, the questions are heavily inspired by the 2018 high school final exam (known as Le Baccalauréat or Le Bac’) in mathematics in France, as can be checked from the above. Including the denomination of powerful number! In Part A, the students had to study the Diophantine equation x²-8y²=1 (E) and show it had an infinite number of (integer) solutions. While these students did not have access to computing facilities, question 1 was solved by sheer enumeration, providing 8=2³ and 9=3² as an answer. While (2) and (3) are rather straightforward, using the prime number decomposition of a and b for (2), the fact that x² and 8y² are powerful for (3). Brute force search leading to x=99,y=35 for the final question.

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