LIX OM - II - Task 1

Determine the maximum possible length of a sequence of consecutive integers, each of which can be expressed in the form $ x^3 + 2y^2 $ for some integers $ x, y $.
A sequence of five consecutive integers -1, 0, 1, 2, 3 satisfies the conditions of the problem: indeed, we have 






On the other hand, among any six consecutive integers, there exists a number, say
$ m $, which gives a remainder of 4 or 6 when divided by 8. The number $ m $ is even; if there were a representation
in the form $ m = x^3 +2y^2 $ for some integers $ x, y $, then the number $ x $ would be even. In this case, however,
we would obtain the divisibility $ 8|x^3 $ and as a result, the numbers $ m $ and $ 2y^2 $ would give the same remainder (4 or 6) when divided by 8.
Therefore, the number $ y^2 $ would give a remainder of 2 or 3 when divided by 4. This is impossible: the equalities






prove that the square of an integer can give a remainder of only 0 or 1 when divided by 4.
We have thus shown that among any six consecutive integers, there exists a number that cannot be represented
in the form $ x^3 +2y^2 $.
Answer: The maximum possible length of such a sequence is 5.