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Fouvry showed that the relative density is positive of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha > \approx .6687$.

Etienne Fouvry, Th ́eoreme de Brun-Titchmarsh; application au th ́eoreme de Fermat, Invent. Math 79 (1985), 383–407. MR0778134 (86g:11052)

show/hide this revision's text 2 Added citation for Fouvry's paper, and "positive density"

Fouvry showed that the relative density is positive of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha > .6687$6687$.

Etienne Fouvry, Th ́eoreme de Brun-Titchmarsh; application au th ́eoreme de Fermat, Invent. Math 79 (1985), 383–407. MR0778134 (86g:11052)

show/hide this revision's text 1

Fouvry showed that the relative density of primes $p$ for which the largest prime factor of $p+a$ is $\ge p^{\alpha}$ for $\alpha > .6687$