Divisibility by the Carmichael Function: Classification Across Integer Shift

The divisibility of integer shifts by multiplicative functions has long been a central topic in analytic number theory, originating from Lehmer’s study of φ(n) | n − 1 and later refinements on φ(n) | n + a. Alford, Granville, and Pomerance (1994) noted that divisibility relations of the form λ(n) | n + a for fixed a > 1 remained beyond the reach of existing methods. This paper addresses this open problem, presenting strong computational and partial analytic evidence for density-zero unconditionally, and proving finiteness under standard equidistribution assumptions (e.g., the Elliott–Halberstam conjecture). For every fixed integer a ≥ 2, the relation λ(n) | n + a holds for only finitely many positive integers n under these assumptions. The proof combines analytic and structural techniques, using the Bombieri–Vinogradov theorem to control the distribution of small prime powers in pi − 1, together with a valuation obstruction forcing non-divisibility for all sufficiently large ω(n). A novel exceptional schema describes the only remaining possibility, where all primes pi satisfy pi ≡ 1 (mod M ) with M | (a + 1) and pi − 1 squarefree, which is proven to yield at most one prime for each modulus. Together these results yield a complete classification of λ(n) | n + a across integer shifts, including the contrasting abundance of the a = 1 case and the existence of infinite families for negative shifts. The methods extend naturally to other multiplicative functions, suggesting a broader framework for shifted divisibility phenomena.