Distribution of Numbers with More than 2 Prime Factors

Theorem

For sufficiently large $x$, there always exists an integer with more than $2$ prime factors between $\paren {x - x^\alpha}$ and $x$, where:

$\alpha \ge 0 \cdotp 477 \ldots$

Proof

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Sources

• 1979: Chen Jingrun: On the distribution of almost primes in an interval II (Sci. Sinica Vol. 22: pp. 253 – 275)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,477 \ldots$