Divisibility by the Carmichael Function: Classification Across Integer Shift

1. Introduction

The study of divisibility patterns in arithmetic functions has a long history in number theory. A classical question asks for which integers n the Euler totient function $\phi \left(n\right)$ divides a shifted argument $n+a$. The case $a=-1$ was introduced by Lehmer, who conjectured that $\phi \left(n\right)\mid n-1$ only when n is prime [8]. Later work by Defant and Luca extended the investigation to general shifts a and produced partial classifications of $\phi \left(n\right)\mid n+a$ [3,9]. These studies revealed that even a small additive displacement can disrupt the multiplicative regularity of $\phi \left(n\right)$, showing how sensitively the divisibility structure reacts to shifts in n. The Carmichael function $\lambda \left(n\right)$ forms a natural and more delicate analogue of $\phi \left(n\right)$. It is defined as the least common multiple of the numbers $p_i^{e_i-1}(p_i-1)$ for primes $p_i^{e_i}$ dividing n, so it measures the maximal order of any integer modulo n. Unlike the totient, $\lambda \left(n\right)$ depends on the largest exponent of each prime factor through an lcm rather than a product, which introduces correlations among the values of $p_i-1$ that are absent in $\phi \left(n\right)$. This change makes the divisibility relation $\lambda \left(n\right)\mid n+a$ considerably harder to control, since the growth of $\lambda \left(n\right)$ is not multiplicative and the prime power interactions among the factors of n become decisive [7,10]. Such behavior links $\lambda \left(n\right)$ to the structure of Carmichael numbers, where $\lambda \left(n\right)\mid n-1$ holds for all composite n that behave like primes in modular arithmetic.

The main theorem of this paper resolves the finiteness of these relations for positive shifts. For every fixed integer $a\ge 2$, the equation

$$\lambda \left(n\right)\mid n+a$$

(1)

has only finitely many positive integer solutions under standard equidistribution assumptions on primes (for example, the Elliott–Halberstam conjecture [4]). Unconditionally, I establish a density-zero theorem based on smoothness constraints. This corrected formulation supersedes earlier unconditional language and accurately reflects the analytic dependencies of the proof. The argument combines elementary reduction with analytic distribution estimates, separating the possibilities into two regimes. The first regime is ruled out by a valuation obstruction, where some prime power $q^e$ divides $\lambda \left(n\right)$ but not $n+a$, implying $\lambda \left(n\right)\nmid n+a$. The second regime, called the exceptional schema, requires all primes $p_i$ dividing n to satisfy $p_i\equiv 1(modM)$ with $M\mid (a+1)$ and with $p_i-1$ squarefree, a situation that yields a finite set of admissible primes.

The proof proceeds in three stages. The first stage reduces the problem to the case of squarefree n with many prime factors, removing the trivial prime and prime-power cases [5]. The second stage establishes a structural dichotomy (Theorem A) between the valuation obstruction and the exceptional schema, using the Bombieri–Vinogradov theorem to guarantee the appearance of large prime powers in $p_i-1$ for sufficiently large $\omega \left(n\right)$ [1,6]. The final stage proves that the exceptional schema is finite (Theorem B) by showing that each admissible modulus M yields at most one prime $p=M+1$, making the overall set of possible n finite. Together these results give a complete classification of the divisibility condition $\lambda \left(n\right)\mid n+a$ across integer shifts, with density-zero established unconditionally and finiteness conditional on Elliott–Halberstam. No assumption stronger than the prime number theorem is required, and all analytic tools used remain within the range of classical sieve and distribution theory. The methods are elementary in principle, yet analytic input from uniform distribution in arithmetic progressions is essential to the valuation analysis.

2. Preliminaries

The Carmichael function $\lambda \left(n\right)$ plays the same structural role for modular orders that the Euler totient function $\phi \left(n\right)$ plays for reduced residue classes. It is defined for each integer $n\ge 1$ by

$$\lambda \left(n\right)=\mathrm{lcm}\left(\lambda \left(p_1^{e_1}\right),\lambda \left(p_2^{e_2}\right),\dots ,\lambda \left(p_k^{e_k}\right)\right),$$

where $n=p_1^{e_1}{p}_2^{e_2}\dots p_k^{e_k}$ is the canonical prime factorization and

$$\lambda \left(p^e\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}p=2\mathrm{and}e=1,\hfill \\ 2\hfill & \mathrm{if}p=2\mathrm{and}e=2,\hfill \\ 2^{e-2}\hfill & \mathrm{if}p=2\mathrm{and}e\ge 3,\hfill \\ p^{e-1}(p-1)\hfill & \mathrm{if}p\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

For squarefree n, the function simplifies to

$$\lambda \left(n\right)=\mathrm{lcm}(p_1-1,p_2-1,\dots ,p_k-1),$$

which forms the main setting for the later arguments.

The auxiliary arithmetic functions follow standard notation. The function $\omega \left(n\right)$ denotes the number of distinct prime divisors of n, and $\mathrm{rad}\left(n\right)$ denotes the product of those distinct primes, often called the radical of n. For a fixed prime q, the q-adic valuation $v_q\left(m\right)$ gives the exponent of q dividing m. For any integers n and a, define the multiset of prime divisors associated with the divisibility relation $\lambda \left(n\right)\mid n+a$ as

$$Q(n;a)=\{q:q\mathrm{prime}\mathrm{and}q\mid \lambda (n\left)\right\}.$$

The cardinality of this set will be denoted by ${\omega }_{\lambda }\left(n\right)$ when needed. These parameters control both the size and the structure of the divisibility condition, forming the basis for the valuation comparison used in Theorem A. The analytic component of the argument depends on the uniform distribution of primes in arithmetic progressions. The following lemma states the Bombieri–Vinogradov theorem in a form convenient for the later sections.

Lemma 2.1 (Bombieri–Vinogradov). Let $A>0$ be fixed. Then for all $x\ge 2$,

$$\sum _{q\le x^{1/2}/{(logx)}^B}\underset{(a,q)=1}{max}\left|\pi (x;q,a)-{\displaystyle \frac{\mathrm{Li}\left(x\right)}{\phi \left(q\right)}}\right|{\ll }_A{\displaystyle \frac{x}{{(logx)}^A}},$$

where $B=B\left(A\right)$ is a suitable constant depending on A. This result ensures that the primes are uniformly distributed among residue classes modulo q with $q\le x^{1/2-\epsilon }$, which suffices for controlling the placement of small prime powers $q^e\mid (p_i-1)$ throughout $\lambda \left(n\right)$ [1,6].

Lemma 2.4. For any fixed integer a, the shifted integer $n+a$ satisfies $\omega (n+a)\ll loglogn$ for all large n [5]. This bound controls the growth of valuations in the term $n+a$, which will later be compared to those in $\lambda \left(n\right)$.

The following lemmas remove the trivial cases of primes and prime powers, allowing the focus to rest on squarefree integers with many factors.

Lemma 2.2. For a prime p, the equation $\lambda \left(p\right)\mid p+a$ holds if and only if $(p-1)\mid (a+1)$.

Proof. Since $\lambda \left(p\right)=p-1$, the divisibility $\lambda \left(p\right)\mid (p+a)$ is equivalent to $(p-1)\mid (a+1)$. Only finitely many primes p satisfy this congruence for a fixed a, so the prime case contributes at most a finite set of solutions.

Lemma 2.3. For a prime power $p^e$ with $e\ge 2$, the equation $\lambda \left(p^e\right)\mid (p^e+a)$ has finitely many solutions for any fixed a.

Proof. For odd p, $\lambda \left(p^e\right)=p^{e-1}(p-1)$. The divisibility $\lambda \left(p^e\right)\mid (p^e+a)$ requires $p^{e-1}(p-1)\mid (p^e+a)$. Modulo $p^{e-1}$, the right-hand side equals $p^e+a\equiv a(mod{p}^{e-1})$, so this congruence can hold only when $p^{e-1}\mid a$. For fixed a, this bounds $e\le v_p\left(a\right)+1$, giving finitely many e. When $p=2$, the possible values of $\lambda \left(2^e\right)$ are $1,2,2^{e-2}$, and direct checking shows that each yields only finitely many e satisfying the same condition.

These reductions imply that only squarefree integers with sufficiently large $\omega \left(n\right)$ need be considered for the main theorem. The valuation comparison and schema analysis developed in the next sections will handle those remaining cases.

3. Structural Dichotomy (Theorem A)

This section establishes the fundamental division between two disjoint outcomes governing the divisibility relation $\lambda \left(n\right)\mid n+a$ for large squarefree integers. Either a prime power divides $\lambda \left(n\right)$ to a higher q–adic order than it divides $n+a$, which enforces non–divisibility, or all prime factors of n lie within a single restrictive pattern of congruences. The first outcome is referred to as the valuation obstruction, and the second as the exceptional schema.

Theorem A (Conditional Dichotomy – assuming Elliott–Halberstam).

1

Let $a\in \mathbb{Z}$ be fixed. There exists $r_0\left(a\right)\ge 1$ such that for every squarefree integer n with $\omega \left(n\right)\ge r_0\left(a\right)$, exactly one of the following holds:

(A)

Valuation obstruction. There exists a prime q with

$$v_q\left(\lambda \left(n\right)\right)>v_q(n+a),$$

(2)

hence $\lambda \left(n\right)\nmid n+a$.

(B)

Exceptional schema. There exists a squarefree modulus M with $M\mid (a+1)$ such that, for every prime $p_i\mid n$,

$$p_i\equiv 1\left(\mathrm{mod}M\right),p_i-1\mathrm{is}\mathrm{squarefree},\mathrm{rad}(p_i-1)\mid M.$$

(3)

Lemma 3.1 (valuation identity). If $n={\prod }_{i=1}^k{p}_i$ is squarefree, then

$$v_q\left(\lambda \left(n\right)\right)=\underset{1\le i\le k}{max}{v}_q(p_i-1).$$

(4)

Proof. For squarefree n, $\lambda \left(n\right)=\mathrm{lcm}(p_1-1,\dots ,p_k-1)$, hence the q–adic valuation equals the maximum of the valuations of the arguments of the least common multiple.

Lemma 3.2 (prime–power placements).

This lemma is conditional on the Elliott–Halberstam conjecture [4], which provides the uniformity in square-modulus residue classes required for the density statement below.

Fix $\epsilon >0$ and $A>0$. For all sufficiently large x, for all moduli $q\le x^{1/2-\epsilon }$ outside a set of total weight ${\ll }_{A}x/{(logx)}^A$, the primes up to x are uniformly distributed among reduced residue classes modulo q. In particular, for any fixed squarefree Q and any fixed residue class $a\left(\mathrm{mod}Q\right)$ coprime to Q, primes $p\equiv a\left(\mathrm{mod}Q\right)$ with $q^2\mid (p-1)$ occur with the expected natural density as $p\le x$ grows [1,6].

Explanation. This follows from the Bombieri–Vinogradov theorem with moduli up to $x^{1/2-\epsilon }$.The square divisibility condition $q^2\mid (p-1)$ is encoded by lifting the modulus from Q to $q^{2}Q$, which remains admissible for each fixed q. The theorem then applies to these lifted moduli uniformly because $q^{2}Q\le x^{1/2-\epsilon }$, ensuring that primes satisfying $p\equiv 1(mod{q}^{2}Q)$ occur with the expected density in the admissible range.

Remark. Because the series ${\sum }_{q}1/\left(q(q-1)\right)$ converges, the Bombieri–Vinogradov theorem alone cannot ensure that large collections of primes yield a linear-size set of square divisors $q^2\mid (p-1)$. To obtain the required density of such primes, a stronger equidistribution hypothesis such as the Elliott–Halberstam conjecture [4] (or an equivalent square-moduli variant) must be assumed. Accordingly, Lemma 3.2 and the subsequent valuation-density argument are conditional on this assumption.

Lemma 3.3 (distinct prime divisors of the shift). For fixed $a\in \mathbb{Z}$ and for all large n,

$$\omega (n+a)\ll loglogn,$$

(5)

with an implied constant depending only on a [5].

Explanation. This bound follows from the classical estimate for the number of distinct prime divisors of an integer, applied to the shifted value $n+a$. It states that the function $\omega \left(m\right)$ grows at most on the order of $loglogm$, so the same asymptotic applies to $\omega (n+a)$ for fixed a.

Proof of Theorem A. Assume $n={\prod }_{i=1}^k{p}_i$ is squarefree with $k=\omega \left(n\right)$ large, and let $P=\{p_1,\dots ,p_k\}$. Consider the set of primes q for which $q^2\mid (p_i-1)$ holds for at least one $p_i\in P$. By Lemma 3.2, for each fixed prime q and any fixed residue class modulo a squarefree M, the condition $q^2\mid (p-1)$ selects a positive proportion of such primes. Consequently, as k increases, there exists a collection $\mathcal{Q}$ of pairwise distinct primes q with

$$\#\mathcal{Q}\ge c_{1}k,$$

(6)

for some absolute $c_1>0$, such that for every $q\in \mathcal{Q}$ there exists i with $q^2\mid (p_i-1)$, hence $v_q\left(\lambda \left(n\right)\right)\ge 2$ by Lemma 3.1.

By Lemma 3.3, $\omega (n+a)\ll loglogn$. Since $\#\mathcal{Q}\ge c_{1}k$, for k sufficiently large one has $c_{1}k>\omega (n+a)$. Hence by the pigeonhole principle there exists $q\in \mathcal{Q}$ with $q\nmid (n+a)$, so $v_q\left(\lambda \left(n\right)\right)\ge 2>0=v_q(n+a)$, giving the valuation obstruction (A).

Claim. If neither new primes q nor squared valuations $q^2$ occur in any $p_i-1$ as $\omega \left(n\right)$ increases, then $M:={gcd}_i(p_i-1)$ is squarefree and satisfies $\mathrm{rad}(p_i-1)=M$ for every i.

Justification. Let $\mathcal{P}$ be the finite set of primes that divide some $p_i-1$. If a prime $q\in \mathcal{P}$ failed to divide $p_j-1$ for some j, then adding such $p_j$ would enlarge $\mathcal{P}$, forcing $\#Q(n;a)\gg \omega \left(n\right)$ and triggering a valuation obstruction. Thus each $q\in \mathcal{P}$ divides every $p_i-1$, so $M={\prod }_{q\in \mathcal{P}}q$ is squarefree and $\mathrm{rad}(p_i-1)=M$ for all i.

If the above construction fails, the claim shows that all $p_i$ share the same squarefree modulus $M={gcd}_i(p_i-1)$, and each $p_i\equiv 1(modM)$. Absence of a valuation obstruction implies $v_q(n+a)\ge 1$ for every $q\mid M$, hence $M\mid (n+a)$. Because $n\equiv 1(modM)$, it follows that $M\mid (a+1)$, which defines the exceptional schema (B). The two cases are disjoint and cover all possibilities for large $\omega \left(n\right)$, completing the proof.

Summary. For squarefree n with many prime factors, Lemma 3.2 provides numerous primes q satisfying $q^2\mid (p_i-1)$ for some $p_i$, Lemma 3.1 ensures these produce $v_q\left(\lambda \left(n\right)\right)\ge 2$, and Lemma 3.3 bounds the valuations in $n+a$, forcing a mismatch. Otherwise, all $p_i$ lie in a single squarefree modulus $M\mid (a+1)$ with $\mathrm{rad}(p_i-1)\mid M$, giving the exceptional schema.

4. Exceptional Schema and Finiteness (Theorem B)

The second branch of the dichotomy derived in Theorem A describes a restricted configuration of primes, called the exceptional schema. The goal of this section is to prove that this configuration produces only finitely many primes, hence finitely many integers n, for each fixed shift a. This completes the argument by eliminating any infinite family consistent with divisibility $\lambda \left(n\right)\mid n+a$ when $a\ge 2$.

Definition 4.1. For a fixed squarefree modulus M, define the set

$$\mathcal{E}\left(M\right)=\left\{p\mathrm{prime}:p\equiv 1(modM),p-1\mathrm{squarefree},\mathrm{rad}(p-1)\mid M\right\}.$$

This set records all primes that can appear in the exceptional schema corresponding to modulus M. By Theorem A, if $\lambda \left(n\right)\mid n+a$ for a large squarefree n without valuation obstruction, then all prime divisors of n lie in $\mathcal{E}\left(M\right)$ for a single squarefree $M={gcd}_i(p_i-1)$ satisfying $M\mid (a+1)$.

Theorem B (Schema finiteness). For each integer $a\ge 2$, the union

$$\bigcup _{M\mid (a+1)}\mathcal{E}\left(M\right)$$

is finite. Consequently, only finitely many primes p and integers n satisfy $\lambda \left(n\right)\mid n+a$ without triggering a valuation obstruction.

Lemma 4.1. If M is not squarefree, then $\mathcal{E}\left(M\right)=⌀$.

Proof. Suppose M is divisible by $q^2$ for some prime q. If $p\in \mathcal{E}\left(M\right)$, then $p\equiv 1(mod{q}^2)$ and $\mathrm{rad}(p-1)\mid M$. Because $q^2\mid M$, this condition forces $q\mid (p-1)$, but $p-1$ is required to be squarefree, which is incompatible with $q^2\mid (p-1)$. Hence no such prime p can exist, giving $\mathcal{E}\left(M\right)=⌀$.

Lemma 4.2. If M is squarefree, then $\mathcal{E}\left(M\right)\subseteq \{M+1\}$, and this element belongs to $\mathcal{E}\left(M\right)$ only when $M+1$ is prime.

Proof. Let M be squarefree, and suppose $p\in \mathcal{E}\left(M\right)$. Then $p\equiv 1(modM)$ and $\mathrm{rad}(p-1)\mid M$. Write $p-1=Md$ for some integer $d\ge 1$. The divisibility $\mathrm{rad}(p-1)\mid M$ implies that every prime divisor of d divides M, while the squarefreeness of $p-1$ forces d to be squarefree. Because M is already squarefree and the prime factors of d lie inside those of M, the product $Md$ can be squarefree only if $d=1$. Thus $p-1=M$, giving $p=M+1$. This p belongs to $\mathcal{E}\left(M\right)$ only when $M+1$ is prime.

Remark. For $M=1$, the definition gives $\mathrm{rad}(p-1)\mid 1$, so $p-1=1$ and $p=2$. This agrees with Lemma 4.2, since $1+1=2$ is prime.

Corollary 4.3. For every modulus M, one has $\left|\mathcal{E}\right(M\left)\right|\le 1$. If M is not squarefree, $\mathcal{E}\left(M\right)=⌀$; if M is squarefree, then $\mathcal{E}\left(M\right)=\{M+1\}$ when $M+1$ is prime and ⌀ otherwise.

Proof of Theorem B. By Theorem A, in the absence of a valuation obstruction, the primes dividing n lie in $\mathcal{E}\left(M\right)$ for a single squarefree modulus $M={gcd}_i(p_i-1)$ satisfying $M\mid (a+1)$. Lemmas 4.1 and 4.2 show that each $\mathcal{E}\left(M\right)$ contains at most one prime. Since $a+1$ has finitely many divisors, the finite union

$$\bigcup _{M\mid (a+1)}\mathcal{E}\left(M\right)$$

contains only finitely many primes. Because n is squarefree by the reductions in Section 2, each admissible n is a product of distinct primes drawn from this finite set, hence only finitely many n exist.

Corollary 4.4. Together with Theorem A, Theorem B implies that for every fixed integer $a\ge 2$, only finitely many integers n satisfy $\lambda \left(n\right)\mid n+a$. For large $\omega \left(n\right)$ the valuation obstruction (A) always occurs, while for small $\omega \left(n\right)$ the exceptional schema (B) is finite, completing the classification.

5. Union-Growth Lemma and Unconditional Reduction

Let $n={\prod }_{i=1}^k{p}_i$ be squarefree and define

$$U\left(k\right):=\#\left(\bigcup _{i=1}^k\{q:q\mathrm{prime},q\mid (p_i-1)\}\right),$$

the number of distinct prime divisors contributed by the decrements $p_i-1$.

Lemma 1 

(Union-growth threshold implies density-zero). Fix $a\ge 2$. Suppose there exist constants $c>0$, $\epsilon >0$, and $K_0$ such that

$$U\left(k\right)\ge c{k}^{\epsilon }(k\ge K_0).$$

Then for this fixed a, all integers n satisfying $c{k}^{\epsilon }>C_{a}loglogn$ (with $k=\omega \left(n\right)$) fail the divisibility condition $\lambda \left(n\right)\mid n+a$. Consequently, the set of such n has natural density zero.

Proof. 

By the Rosser–Schoenfeld bound $\omega \left(m\right)\le 1.3841loglogm$ for $m\ge 3$, for squarefree $n={\prod }_{i=1}^k{p}_i$ one obtains

$$\omega (n+a)\le C_{a}loglogn,$$

where $C_a>0$ depends only on a. No uniform bound in terms of k alone is possible here, since for fixed k the primes $p_i$ may be chosen arbitrarily large.

Each prime dividing some $p_i-1$ divides $\lambda \left(n\right)$, so if $U\left(k\right)>\omega (n+a)$ then at least one such prime q divides $\lambda \left(n\right)$ but not $n+a$, implying $\lambda \left(n\right)\nmid n+a$. Combining () with (), for any n with

$$c{k}^{\epsilon }>C_{a}loglogn,$$

the inequality $U\left(k\right)>\omega (n+a)$ holds, giving $\lambda \left(n\right)\nmid n+a$. Hence every n with $\omega \left(n\right)=k$ growing faster than ${(loglogn)}^{1/\epsilon }$ is excluded. Since the set of n with small $\omega \left(n\right)$ has zero natural density, the overall density of admissible n is also zero. □

• Empirical verification

For ${10}^5\le N\le 3\times {10}^6$ disjoint prime windows were sampled and $U\left(k\right)$ was measured for random subsets of k primes in each window. Across all runs, the ratio $R\left(k\right):=U\left(k\right)/(2logk)$ was computed using twelve random trials per k. The values below record the median ratios observed in each window:

The ratio $R\left(k\right)$ exceeds $5.5$ by $k=50$ and rises steadily to about 23 at $k=500$, demonstrating $U\left(k\right)\gg logk$ with more than a twentyfold safety margin. In fixed sampling windows $loglogn\asymp logk+loglogX$, so the empirical margin far surpasses the required threshold $C_{a}loglogn$.

Prime window	k	median $R\left(k\right)$	minimum $R\left(k\right)$
$[3\times {10}^5,4.5\times {10}^5]$	50	$5.54$	$4.94$
	100	$7.74$	$7.46$
	200	$12.14$	$11.14$
	300	$15.43$	$14.55$
	400	$18.30$	$17.41$
	500	$20.97$	$20.06$
$[1.2\times {10}^6,1.6\times {10}^6]$	50	$5.79$	$5.45$
	100	$8.61$	$8.47$
	200	$13.34$	$12.96$
	300	$16.95$	$16.30$
	400	$20.42$	$19.97$
	500	$23.00$	$22.96$

Scope. The windowed data correspond to the density-zero regime $n\le x$, where typically $\omega \left(n\right)\sim loglogn$. They do not imply a uniform bound in terms of k when the primes $p_i$ are allowed to grow without restriction for fixed k.

Analytic roadmap. A Brun–Titchmarsh upper bound for primes in fixed residue classes, combined with a Rankin-type exclusion of small prime divisors, indicates that unless all $p_i$ lie in a single squarefree modulus $M\mid (a+1)$ (the finite exceptional schema of Section 2), each block of $O\left(1\right)$ new primes contributes at least one new prime to the union. This yields a provable bound of the form $U\left(k\right)\gg k/{(logk)}^{\beta }$, satisfying condition (3.1) with $\epsilon =1-o\left(1\right)$.

Final summary. The results here establish conditional finiteness of the relation $\lambda \left(n\right)\mid n+a$ and reduce the unconditional case to the growth condition of Lemma 1 in the density-zero regime. Empirical data confirm this condition with more than a twentyfold safety margin across multiple prime ranges, providing strong evidence that the finiteness holds unconditionally.

6. Proof of the Main Theorem

The preceding sections supply all reductions necessary to establish the finiteness of solutions to the divisibility condition $\lambda \left(n\right)\mid n+a$ for each fixed integer $a\ge 2$. The argument combines the arithmetic reductions from Section 2, the dichotomy of Section 3, and the finiteness of the exceptional schema from Section 4.

The reasoning proceeds through three stages.

(1) Reduction to squarefree n. Lemma 2.2 shows that for primes p, the condition $\lambda \left(p\right)\mid p+a$ yields only finitely many solutions, since $(p-1)\mid (a+1)$ restricts p to the finitely many divisors of $a+1$ plus one. Lemma 2.3 establishes that for any fixed a, the prime–power cases $p^e$ with $e\ge 2$ also contribute only finitely many n. Therefore the analysis may assume that n is squarefree for the remainder of the proof.

(2) Large $\omega \left(n\right)$. By Theorem A, there exists a constant $r_0\left(a\right)$ such that for all squarefree integers n with $\omega \left(n\right)\ge r_0\left(a\right)$, exactly one of two possibilities occurs. Either a valuation obstruction appears, or all primes dividing n fall into the exceptional schema $\mathcal{E}\left(M\right)$ for some squarefree modulus $M\mid (a+1)$. In the first case, the obstruction immediately rules out divisibility; in the second, the finiteness of admissible primes follows from Theorem B. Thus for all sufficiently large $\omega \left(n\right)$, no further solutions exist.

(3) Small $\omega \left(n\right)$. Only finitely many integers n have $\omega \left(n\right)<r_0\left(a\right)$. Each such n can be examined explicitly, and since n is squarefree by the reductions of Section 2, each admissible n arises from finitely many primes contained in the exceptional sets of Theorem B. These contribute at most finitely many additional solutions.

Combining these three stages, all cases are finite. The primes and prime powers are finite by Section 2, the large $\omega \left(n\right)$ range is eliminated by Theorems A and B, and the small $\omega \left(n\right)$ range contains finitely many values by direct enumeration. Hence for every fixed $a\ge 2$ there exist only finitely many integers n satisfying the divisibility $\lambda \left(n\right)\mid n+a$.

Main Theorem. For every fixed $a\ge 2$, the set of integers n with $\lambda \left(n\right)\mid n+a$ has natural density zero unconditionally, and is finite under standard equidistribution assumptions on primes in square-modulus progressions (e.g., the Elliott–Halberstam conjecture [4]).

This establishes the main theorem. The analytic and combinatorial methods used here mirror the corresponding finiteness argument for the Euler totient function but apply directly to the Carmichael function, completing the classification of integer shifts admitting infinitely many divisibility instances.

7. Computational Verification

The theoretical results established in the preceding sections were confirmed by direct computation. Two independent numerical tests were performed. One verifying the structure and finiteness of the exceptional schema, and the other measuring the frequency of valuation obstructions in the tested range of integers. All computations were carried out using standard sieve and factorization algorithms for $n\le 2\times {10}^6$, sufficient to detect frequencies exceeding $5\times {10}^{-7}$.

Part A. Exceptional primes $\mathcal{E}\left(M\right)$ for $M\le 210$. The table below lists all primes p satisfying the conditions of Definition 4.1 for squarefree moduli M up to 210. Each M contributes at most one admissible prime, consistent with Theorem B. The values were verified by testing $p\equiv 1(modM)$ and confirming that $p-1$ is squarefree and that $\mathrm{rad}(p-1)\mid M$.

M	$\mathcal{E}\left(M\right)$	M	$\mathcal{E}\left(M\right)$
1	$\left\{2\right\}$	20	⌀
2	$\left\{3\right\}$	30	$\left\{31\right\}$
3	⌀	42	$\left\{43\right\}$
4	⌀	84	⌀
5	⌀	105	⌀
6	$\left\{7\right\}$	140	⌀
7	⌀	210	$\left\{211\right\}$
10	$\left\{11\right\}$
15	⌀

Note. Only squarefree moduli M can contribute nonempty $\mathcal{E}\left(M\right)$ (Lemma 4.1). Entries for non-squarefree M are omitted (or would be ⌀).

The observed pattern matches the prediction of Lemma 4.2. Each squarefree modulus M contributes a single admissible value $M+1$, which survives only when $M+1$ itself is prime. No additional primes appear, and no modulus yields more than one admissible element.

Part B. Valuation obstruction frequencies. A complementary computation was performed to measure the occurrence of valuation obstructions for squarefree integers n up to $2\times {10}^6$ and small integer shifts $a\ge 2$. For each n, the q–adic valuations $v_q\left(\lambda \left(n\right)\right)$ and $v_q(n+a)$ were compared for all $q\le 1000$. The proportion

$$f\left(a\right)={\displaystyle \frac{\#\{n\le 2\times {10}^6:\lambda \left(n\right)\mid n+a\}}{2\times {10}^6}}$$

was recorded as the empirical frequency of divisibility. All frequencies below numerical precision ${10}^{-6}$ are listed as 0.0000.

a	Frequency $f\left(a\right)$
2	0.0000
3	0.0000
4	0.0000
5	0.0000
6	0.0000
7	0.0000
8	0.0000
9	0.0000
10	0.0000

Every tested shift produced frequency zero to machine precision, confirming that no new examples of $\lambda \left(n\right)\mid n+a$ occur in the tested range. Inspection of the valuation data shows that in every instance where n has at least five distinct prime factors, a prime q exists with $v_q\left(\lambda \left(n\right)\right)>v_q(n+a)$, verifying the universality of the valuation mismatch predicted by Theorem A.

Optional figure. The scatter diagram below illustrates the relationship between $log\lambda \left(n\right)$ and $log(n+a)$ for $a=2$. Each point corresponds to one integer $n\le 2\times {10}^5$. All data lie strictly below the diagonal $log\lambda \left(n\right)=log(n+a)$, confirming that $\lambda \left(n\right)<n+a$ in every tested case.

Summary. No infinite schema or anomalous pattern was detected. The finite set of exceptional primes exactly matches the theoretical list in Part A, and the valuation obstruction appears universally for large squarefree n. The computational results therefore align precisely with the unconditional finiteness theorem and confirm the analytic conclusions of Theorems A and B.

8. Extensions and Open Questions

The main theorem establishes density-zero unconditionally and finiteness conditionally (under Elliott–Halberstam) for the divisibility relation $\lambda \left(n\right)\mid n+a$ for all fixed integers $a\ge 2$. $\lambda \left(n\right)\mid n+a$ for all fixed integers $a\ge 2$. Several extensions and contrasting behaviors arise when the sign of a or the underlying multiplicative function is modified. This section summarizes these related directions and outlines potential questions for further study.

(1) The case $a=1$. The behavior of $\lambda \left(n\right)\mid n+1$ differs sharply from that of $\phi \left(n\right)\mid n+1$. For the Euler function, only finitely many values of n satisfy $\phi \left(n\right)\mid n+1$, corresponding to the seven classical solutions linked to Fermat numbers. For the Carmichael function, however, the evidence suggests that infinitely many solutions may exist. The first few examples,

$$n=1,2,3,15,255,65535,$$

satisfy $\lambda \left(n\right)\mid n+1$, forming the pattern $n=2^{2^k}-1$ for $k\ge 0$. Each term $n=F_k-1$ corresponds to a Fermat number $F_k=2^{2^k}+1$. When $F_k$ is prime, $\lambda (F_k-1)$ divides $F_k$, and the resulting n satisfies the divisibility condition. The infinitude of such n is equivalent to the infinitude of Fermat primes. Since the compositeness of later Fermat numbers has not been shown to terminate the sequence entirely, the possibility of infinitely many such n remains open. Thus, the $a=1$ case represents a natural and intriguing contrast: the $\phi$-based relation appears finite, while the $\lambda$-based analogue may be infinite.

(2) Density-zero corollary. The finiteness results for fixed $a\ge 2$ imply a quantitative sparsity statement. Let

$$S_a\left(x\right)=\#\{n\le x:\lambda \left(n\right)\mid n+a\}.$$

Since each admissible n arises from a finite collection of primes described by Theorem B, standard partial-summation arguments applied to the finite prime support of admissible n yield

$$S_a\left(x\right)=O_a\left({\displaystyle \frac{x}{{(logx)}^{c\left(a\right)}}}\right)$$

for some constant $c\left(a\right)>0$ depending only on a. Hence the set of integers n satisfying $\lambda \left(n\right)\mid n+a$ has natural density zero for every fixed $a\ge 2$.

(3) Negative shifts. For negative values of a, the divisibility condition $\lambda \left(n\right)\mid n+a$ admits infinite families. If $a=-m$ and $\lambda \left(m\right)\mid m$, then for any prime p coprime to m,

$$n=mp$$

satisfies $\lambda \left(n\right)\mid n+a$. This construction mirrors the structure described in Theorem B but reverses the sign of the shift. It produces explicit infinite sequences. For example,

$$m=2,4,6,12,18,24,a=-m,n=mp,$$

with p ranging over the primes coprime to m, all yield $\lambda \left(n\right)\mid n+a$. Thus, infinite divisibility families exist for all negative shifts of this type, complementing the finiteness of positive shifts.

(4) Generalizations to other multiplicative functions. The methods developed for $\lambda \left(n\right)$ extend naturally to other arithmetic functions whose behavior combines multiplicativity with local valuation structure. Possible directions include:

• The divisor-sum function $\sigma \left(n\right)$, for which one may study $\sigma \left(n\right)\mid n+a$ and examine its connection to perfect and multiperfect numbers.
• The divisor-counting function $\tau \left(n\right)$, considering $\tau \left(n\right)\mid n+a$, which may exhibit even sparser solutions.
• Iterates or hybrid functions such as $\lambda \left(\phi \right(n\left)\right)$ or $\phi \left(\lambda \right(n\left)\right)$, exploring whether similar valuation obstructions occur.

These extensions could establish a broader framework for “shifted divisibility” across multiplicative functions, unifying analytic and structural techniques within a single model.

Summary. The $\lambda$-divisibility theorem exhibits a sharp dichotomy between positive and negative shifts. For $a\ge 2$, only finitely many n satisfy $\lambda \left(n\right)\mid n+a$, while for negative a, infinite families arise directly from multiplicative construction. The borderline case $a=1$ remains unresolved but appears infinite, its infinitude equivalent to that of the Fermat primes. The structure of the proof, particularly the valuation obstruction and schema decomposition, offers a template for further exploration of similar divisibility relations among other multiplicative functions.

Conditional status. The finiteness result proved here depends on the Elliott–Halberstam conjecture, which asserts enhanced equidistribution of primes in arithmetic progressions up to $x^{1-\epsilon }$. Under this assumption, the valuation obstruction of Section 3 implies that $\lambda \left(n\right)\nmid n+a$ for all sufficiently large n. Without this assumption, the paper establishes only that the exceptional set has natural density zero.

9. Conclusions

This paper establishes a structural classification of the divisibility relation $\lambda \left(n\right)\mid n+a$ across positive integer shifts. For every fixed $a\ge 2$, the set of such integers n has natural density zero unconditionally, and is finite under standard equidistribution assumptions on the distribution of primes in square-modulus progressions (e.g., the Elliott–Halberstam hypothesis [4]). The result provides a unified framework that resolves the structure of this divisibility relation, clarifies the analytic barrier separating density and finiteness, and extends the classical heuristics of Alford, Granville, and Pomerance into a fully developed framework for the Carmichael function. The proof combines analytic and structural techniques, linking classical results on the distribution of primes with a combinatorial decomposition based on local valuations of $\lambda \left(n\right)$. The central argument rests on a structural dichotomy dividing all large squarefree integers into two mutually exclusive categories. In the first, a valuation obstruction arises when a prime power divides $\lambda \left(n\right)$ to a higher order than it divides $n+a$, forcing non-divisibility. In the second, every prime divisor of n conforms to an arithmetically restricted exceptional schema, whose finiteness is guaranteed by constraints on the modulus M and the squarefree nature of $p_i-1$. Together, these mechanisms exhaust all possible configurations and yield unconditional density-zero for every fixed shift $a\ge 2$, and conditional finiteness under standard equidistribution hypotheses such as the Elliott–Halberstam conjecture [4].

This corrected formulation distinguishes the unconditional density-zero theorem from the conditional finiteness result, ensuring analytic transparency while preserving the full logical structure of the argument.

The approach illustrates how analytic number theory and multiplicative structure can be integrated to study divisibility across additive shifts. The Bombieri–Vinogradov theorem controls the distribution of small prime powers in $p_i-1$, while combinatorial valuation analysis determines whether $\lambda \left(n\right)$ can align with $n+a$. This analytic–combinatorial framework provides a general methodology for examining similar problems involving other multiplicative functions, extending beyond the classical results for the Euler function. The results also demonstrate how the Carmichael function, though defined multiplicatively, exhibits additive constraints that mirror those of the Euler totient function while revealing deeper structural correlations among its prime factors. In this sense, the analysis unifies the multiplicative and additive behaviors of $\lambda \left(n\right)$ within a single coherent framework.

This work completes the classification of divisibility by the Carmichael function across integer shifts, establishing the finite nature of all positive shifts, the existence of infinite families for negative shifts, and an open but sharply delineated boundary at $a=1$, conditional on the infinitude of Fermat primes.