On the largest prime factors of $p-1$ and $p+1$

Source abstract

We prove that each of the inequalities $P^{+}(p+1)>P^{+}(p-1)$ and $P^{+}(p-1)>P^{+}(p+1)$ holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.