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Pomerance conjecture on the density of smooth shifted primes

Establish that π(x, y)/π(x) ∼ Ψ(x, y)/x as y → ∞, where π(x, y) counts primes p ≤ x with P⁺(p − 1) ≤ y (i.e., p − 1 is y-smooth) and Ψ(x, y) counts integers ≤ x all of whose prime factors are ≤ y.

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Background

The conjecture asserts that the density of y-smooth shifted primes p − 1 among shifted primes mirrors the density of y-smooth integers among all integers, suggesting a strong regularity in the distribution of smooth values of p − 1.

The paper shows that the Adleman–Pomerance–Rumely maximal-order conjecture would follow from this smooth shifted primes conjecture, highlighting its significance in understanding extremal behavior of ω*(n).

References

Indeed, Pomerance has conjectured that if $x\ge y\ge1$, then \begin{equation}\label{eq:smooth p-1} \frac{\pi(x,y)}{\pi(x)}\sim\frac{\Psi(x,y)}{x} \end{equation} as $y\to\infty$.

eq:smooth p-1:

π(x,y)π(x)∼Ψ(x,y)x\frac{\pi(x,y)}{\pi(x)}\sim\frac{\Psi(x,y)}{x}

— The maximal order of the shifted-prime divisor function (2510.14167 - Fan et al., 15 Oct 2025) in Concluding remarks, Equation (smooth p-1)