grandes-ecoles 2025 Q19

grandes-ecoles · France · centrale-maths1__official Number Theory Prime Counting and Distribution

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$. Deduce that there exists a non-zero natural integer $N$ such that, for all $n \geqslant N , d _ { n } \leqslant 3 ^ { n }$.
One may use the prime number theorem: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$. Deduce that there exists a non-zero natural integer $N$ such that, for all $n \geqslant N , d _ { n } \leqslant 3 ^ { n }$.

One may use the prime number theorem: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.

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