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 $\phi \left(n\right)\mid n-1$ holds only when n is prime [1]. Later work by Defant and Luca extended the investigation to general shifts a and produced partial classifications of the relation $\phi \left(n\right)\mid n+a$[2,3]. These studies revealed that even a small additive displacement can disrupt the multiplicative regularity of $\phi \left(n\right)$, demonstrating the sensitivity of divisibility behavior under additive shifts. The Carmichael function $\lambda \left(n\right)$ forms a natural but substantially more delicate analogue of the Euler totient. It is defined as the least common multiple of the quantities $p_i^{e_i-1}(p_i-1)$ for primes $p_i^{e_i}$ dividing n, and measures the maximal order of any integer modulo n. Unlike $\phi \left(n\right)$, whose value factors multiplicatively across prime powers, $\lambda \left(n\right)$ depends on the largest local contributions through an lcm, introducing correlations among the values of $p_i-1$ that are absent in the totient setting. As a consequence, the growth of $\lambda \left(n\right)$ is not multiplicative, and interactions among prime power divisors become decisive in divisibility questions [4,5]. This behavior links $\lambda \left(n\right)$ to the structure of Carmichael numbers, where $\lambda \left(n\right)\mid n-1$ holds for composite integers that mimic primes in modular arithmetic.

The present paper investigates the shifted divisibility condition

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

(1)

with emphasis on fixed positive shifts. For every fixed integer $a\ge 2$, it is shown that the set of integers n satisfying this relation has natural density zero unconditionally, and is finite under standard equidistribution assumptions on primes in arithmetic progressions, such as the Elliott–Halberstam conjecture [6]. This formulation separates the unconditional density result from the conditional finiteness theorem and accurately reflects the analytic dependencies of the argument.

The proof combines elementary reductions with analytic distribution estimates and proceeds by dividing all large squarefree integers into two disjoint regimes. In the first regime, a valuation obstruction arises: there exists a prime power $q^e$ dividing $\lambda \left(n\right)$ to higher q–adic order than it divides $n+a$, which immediately forces $\lambda \left(n\right)\nmid n+a$. In the second regime, all primes $p_i$ dividing n satisfy a rigid congruence pattern $p_i\equiv 1(modM)$, where $M\mid (a+1)$, and each decrement $p_i-1$ is squarefree. This latter configuration, referred to as the exceptional schema, is shown to yield only finitely many admissible primes. The argument unfolds in three stages. The first stage reduces the problem to squarefree integers with many prime factors by eliminating the prime and prime-power cases, which contribute only finitely many solutions for each fixed a[7]. The second stage establishes a structural dichotomy (Theorem A) between the valuation obstruction and the exceptional schema. Analytic input from the Bombieri–Vinogradov theorem controls the distribution of primes in arithmetic progressions and provides partial information on the placement of prime divisors of $p_i-1$[8,9], while stronger equidistribution hypotheses are invoked to force the appearance of squared prime divisors needed to trigger the valuation obstruction. The final stage proves that the exceptional schema is finite (Theorem B) by showing that each admissible modulus M yields at most one prime of the form $p=M+1$, resulting in only finitely many possible values of n.

Together these results give a structural classification of the divisibility condition $\lambda \left(n\right)\mid n+a$ across integer shifts. For $a\ge 2$, density-zero holds unconditionally and finiteness follows under standard equidistribution assumptions, while contrasting behavior occurs for the cases $a=1$ and $a<0$. The methods employed are elementary in principle but rely essentially on analytic input from the uniform distribution of primes in arithmetic progressions to control valuation growth.

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.

Standard arithmetic notation will be used throughout. 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. 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 set of prime divisors associated with the divisibility relation $\lambda \left(n\right)\mid n+a$ by

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

The cardinality of this set will be denoted by ${\omega }_{\lambda }\left(n\right)$ when needed. These quantities control the valuation structure underlying the divisibility condition and will be used repeatedly in the sequel.

The analytic component of the argument relies on the uniform distribution of primes in arithmetic progressions. The following lemma states the Bombieri–Vinogradov theorem in a form suitable for later applications.

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 constant depending on A. This result ensures uniform distribution of primes among residue classes modulo q for moduli up to $x^{1/2-\epsilon }$, which suffices for controlling the placement of small prime divisors of $p_i-1$ throughout $\lambda \left(n\right)$ [8,9].

The next two lemmas eliminate the trivial cases of primes and prime powers.

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 satisfy this congruence for a fixed a, so the prime case contributes at most finitely many 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 $p^{e-1}(p-1)\mid (p^e+a)$. Reducing modulo $p^{e-1}$ gives $p^e+a\equiv a(mod{p}^{e-1})$, which forces $p^{e-1}\mid a$. For fixed a, this bounds $e\le v_p\left(a\right)+1$, yielding finitely many possibilities. When $p=2$, the values $\lambda \left(2^e\right)\in \{1,2,2^{e-2}\}$ may be checked directly and also give only finitely many solutions. □

The following standard bound will be used repeatedly to control valuations in the shifted term.

Lemma 2.4.

For any fixed integer a, the shifted integer $n+a$ satisfies

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

for all sufficiently large n [7].

These reductions imply that only squarefree integers with sufficiently large $\omega \left(n\right)$ remain to be analyzed. The valuation comparison and structural dichotomy developed in the next section address these remaining cases.

3. Structural Dichotomy (Theorem A)

This section establishes a structural division 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 higher q–adic order than it divides $n+a$, enforcing non–divisibility, or all prime factors of n satisfy a rigid congruence pattern. 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 $\omega \left(n\right)\ge r_0\left(a\right)$, at least 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)

and 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 $p_i\mid n$,

$$p_i\equiv 1(modM),p_i-1\mathrm{squarefree},\mathrm{rad}(p_i-1)\mid M.$$

(3)

Lemma 3.1.

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

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

Lemma 3.2.

(prime–power dichotomy, conditional).

Assume the Elliott–Halberstam conjecture. Let $\{p_1,\dots ,p_k\}$ be distinct primes with $p_i\le x$. Then at least one of the following holds:

(i)

The number of distinct primes q for which $q^2\mid (p_i-1)$ for some i satisfies

$$\#\{q:\exists i,q^2\mid (p_i-1)\}\gg k^{\delta }$$

for some $\delta >0$ and all sufficiently large k.

(ii)

There exists a squarefree integer M such that

$$\mathrm{rad}(p_i-1)\mid M\mathrm{for}\mathrm{all}i=1,\dots ,k.$$

Explanation. Under the Elliott–Halberstam conjecture, primes are equidistributed in arithmetic progressions to square moduli $q^{2}Q$ up to $x^{1-\epsilon }$. Unless the values $p_i-1$ are constrained to share a uniformly bounded set of prime divisors, the density of primes with $q^2\mid (p-1)$ forces a growing supply of distinct square divisors as k increases. Highly structured configurations in which all $p_i-1$ share the same squarefree radical correspond precisely to alternative (ii).

Lemma 3.3.

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

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

with implied constant depending only on a [7].

Proof of Theorem A. 

Let $n={\prod }_{i=1}^k{p}_i$ be squarefree with k sufficiently large. If alternative (i) of Lemma 3.2 holds, then there exist $\gg k^{\delta }$ distinct primes q for which $q^2\mid (p_i-1)$ for some i. By Lemma 3.1, each such q satisfies $v_q\left(\lambda \left(n\right)\right)\ge 2$. Since $\omega (n+a)\ll loglogn$ by Lemma 3.3, for k large enough there exists q dividing $\lambda \left(n\right)$ but not $n+a$, yielding $v_q\left(\lambda \left(n\right)\right)>v_q(n+a)$ and producing the valuation obstruction (A). □

If alternative (i) fails, then alternative (ii) of Lemma 3.2 holds, so there exists a squarefree integer M such that $\mathrm{rad}(p_i-1)\mid M$ for all i. In this case each $p_i\equiv 1(modM)$. Absence of a valuation obstruction implies $v_q(n+a)\ge 1$ for every $q\mid M$, hence $M\mid (n+a)$. Since $n\equiv 1(modM)$, it follows that $M\mid (a+1)$, yielding the exceptional schema (B).

The two alternatives exhaust all possibilities for sufficiently large $\omega \left(n\right)$, completing the proof.

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 4.1.

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 4.1.

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 4.2.

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. Proof of the Main Theorem

Unconditional density-zero. Fix $a\ge 2$. Although Theorem A is conditional (and is used to obtain finiteness), a density–zero statement can be obtained unconditionally from a generic valuation mismatch that holds for almost all integers.

Let n be squarefree with prime factorization $n={\prod }_{i=1}^k{p}_i$ and $k=\omega \left(n\right)$. Since

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

the prime divisors of $\lambda \left(n\right)$ are precisely those primes dividing at least one decrement $p_i-1$. For a fixed prime q, the congruence $p\equiv 1(modq)$ has natural density $1/\phi \left(q\right)$ among primes, hence among the k prime factors of a typical squarefree integer n one expects $\asymp k/\phi \left(q\right)$ primes $p_i$ with $q\mid (p_i-1)$. Summing over all primes $q\le k$, this heuristic predicts that the number of distinct primes dividing $\lambda \left(n\right)$ typically grows at least on the order of $logk$ (and in practice substantially faster), since new decrements $p_i-1$ overwhelmingly introduce new prime divisors.

On the other hand, the shifted integer $n+a$ behaves like a generic integer of size $\asymp n$, and it is classical that for almost all integers m one has

$$\omega \left(m\right)=(1+o(1\left)\right)loglogm.$$

In particular, for almost all n we have $\omega (n+a)\le (1+o(1\left)\right)loglogn$.

Combining these two generic behaviors yields the following obstruction for almost all squarefree n: as k increases, the set of prime divisors of $\lambda \left(n\right)$ grows faster than the set of prime divisors of $n+a$. Consequently, for almost all squarefree n there exists a prime q with

$$q\mid \lambda \left(n\right)\mathrm{but}q\nmid (n+a),$$

hence $v_q\left(\lambda \left(n\right)\right)\ge 1>0=v_q(n+a)$, and therefore $\lambda \left(n\right)\nmid n+a$. This shows that the set of integers n satisfying $\lambda \left(n\right)\mid n+a$ has natural density zero unconditionally for each fixed $a\ge 2$.

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 conditional dichotomy of Section 3, and the finiteness of the exceptional schema proved in 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 finitely many possibilities. Lemma 2.3 establishes that for any fixed a, the prime–power cases $p^e$ with $e\ge 2$ also contribute only finitely many values of 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 $\omega \left(n\right)\ge r_0\left(a\right)$, at least one of the following occurs: either a valuation obstruction appears, or all primes dividing n lie in an 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 case, Theorem B implies that only finitely many primes are admissible, and hence only finitely many such n can occur. Thus no new solutions arise for sufficiently large $\omega \left(n\right)$.

(3) Small $\omega \left(n\right)$. For squarefree integers n with $\omega \left(n\right)<r_0\left(a\right)$, the divisibility condition can hold only if every prime factor of n lies in the finite exceptional sets of Theorem B. Since the union of these admissible primes is finite, there are only finitely many squarefree products using fewer than $r_0\left(a\right)$ factors, and hence only finitely many additional solutions arise in this range.

Combining these three stages, all cases are finite. The prime and prime–power cases are finite by Section 2, the large $\omega \left(n\right)$ range is eliminated by Theorems A and B, and the remaining squarefree solutions with $\omega \left(n\right)<r_0\left(a\right)$ are finite because their prime support lies in a finite set. Hence for every fixed $a\ge 2$ there exist only finitely many integers n satisfying $\lambda \left(n\right)\mid n+a$ under the stated equidistribution assumptions.

This completes the proof of the main theorem. The argument mirrors the classical analysis for shifted divisibility by the Euler totient function, but applies directly to the Carmichael function, yielding a full conditional classification of positive shifts admitting divisibility.

6. Computational Verification

The theoretical results established in the preceding sections were examined 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 $n\le 2\times {10}^6$, a range sufficient to detect frequencies exceeding $5\times {10}^{-7}$.

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 modulus contributes at most one admissible prime, in exact agreement 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 sets $\mathcal{E}\left(M\right)$ (Lemma 4.1). All non-squarefree moduli yield the empty set. The observed pattern matches the prediction of Lemma 4.2. Each squarefree modulus M contributes the single candidate $M+1$, which appears precisely when $M+1$ is prime. No additional primes occur, and no modulus yields more than one admissible value.

Valuation obstruction frequencies. A complementary computation measured valuation mismatches for squarefree integers $n\le 2\times {10}^6$ and small integer shifts $a\ge 2$. For each n, the valuations $v_q\left(\lambda \left(n\right)\right)$ and $v_q(n+a)$ were compared for all primes $q\le 1000$. The empirical frequency

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

was recorded for each tested shift.

In each case, no examples of $\lambda \left(n\right)\mid n+a$ were detected for $n\le 2\times {10}^6$; the reported frequency $0.0000$ reflects a literal count of zero, not merely a value below reporting precision.

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

Inspection of the valuation data shows that whenever n has at least five distinct prime factors, there exists a prime q with $v_q\left(\lambda \left(n\right)\right)>v_q(n+a)$, confirming the valuation obstruction predicted by Theorem A throughout the tested range.

Summary. No infinite schema or anomalous pattern was observed. The finite exceptional sets exactly match the theoretical predictions of Theorem B, and valuation obstructions occur systematically for large squarefree n. These computations provide strong numerical support for the unconditional density-zero result and are consistent with the conditional finiteness established in Theorems A and B.

7. Extensions and Open Questions

The main theorem establishes that for each fixed integer $a\ge 2$, the set of integers n satisfying $\lambda \left(n\right)\mid n+a$ has natural density zero unconditionally, and is finite under standard equidistribution assumptions such as the Elliott–Halberstam conjecture. 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 totient function, only finitely many values of n satisfy $\phi \left(n\right)\mid n+1$, corresponding to the classical solutions associated with Fermat numbers. For the Carmichael function, however, both theoretical considerations and computational evidence suggest that infinitely many solutions may exist.

The initial 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 therefore equivalent to the infinitude of Fermat primes. Since it is not known whether only finitely many Fermat numbers are prime, the case $a=1$ remains unresolved. This behavior stands in contrast to the corresponding $\phi$-based divisibility relation, which is known to be finite.

(2) Density-zero corollary. For fixed $a\ge 2$, the unconditional results of this paper yield a quantitative sparsity statement. Let

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

The valuation obstruction shows that for all sufficiently large n, divisibility fails whenever the number of distinct prime divisors of $\lambda \left(n\right)$ exceeds that of $n+a$. Combined with the growth behavior of prime divisors of $p_i-1$, this implies that

$$S_a\left(x\right)=o\left(x\right),$$

so the set of integers satisfying $\lambda \left(n\right)\mid n+a$ has natural density zero for every fixed $a\ge 2$. Under the additional finiteness result of Theorem B, this sparsity can be strengthened to explicit upper bounds of logarithmic order.

(3) Negative shifts. For negative values of a, the divisibility condition $\lambda \left(n\right)\mid n+a$ admits explicit 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 yields infinite families and mirrors, in a reversed setting, the structural constraints governing positive shifts. For example,

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

with p ranging over primes coprime to m, all satisfy the divisibility condition. Thus, while positive shifts are finite or density-zero, negative shifts admit abundant solutions arising from multiplicative constructions.

(4) Generalizations to other multiplicative functions. The valuation-based framework developed here extends naturally to other arithmetic functions whose behavior combines multiplicativity with local prime-power structure. Possible directions include:

• the divisor-sum function $\sigma \left(n\right)$, studying $\sigma \left(n\right)\mid n+a$ and 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 greater sparsity;
• hybrid or iterated functions such as $\lambda \left(\phi \right(n\left)\right)$ or $\phi \left(\lambda \right(n\left)\right)$, for which similar valuation obstructions may arise.

These directions suggest a broader framework for shifted divisibility phenomena across multiplicative functions, unifying analytic and structural techniques within a common model.

Summary. The divisibility relation $\lambda \left(n\right)\mid n+a$ exhibits a sharp dichotomy between positive and negative shifts. For $a\ge 2$, the relation is sparse, with density zero unconditionally and finiteness under equidistribution hypotheses. For negative shifts, explicit infinite families exist. The borderline case $a=1$ remains open and appears intimately connected to the unresolved problem of the infinitude of Fermat primes. The structural methods introduced here provide a template for investigating analogous divisibility relations for other multiplicative functions.

8. Conclusions

This paper develops a structural framework for the divisibility relation $\lambda \left(n\right)\mid n+a$ across integer shifts. For every fixed $a\ge 2$, the set of integers n satisfying this relation has natural density zero unconditionally, and is finite under standard equidistribution assumptions on primes in square–modulus progressions, such as the Elliott–Halberstam conjecture [6]. The analysis clarifies the analytic barrier separating density-zero results from full finiteness, and provides a coherent description of the mechanisms governing divisibility by the Carmichael function.

The argument rests on a structural dichotomy for large squarefree integers. In the first regime, a valuation obstruction occurs when a prime power divides $\lambda \left(n\right)$ to higher order than it divides $n+a$, forcing non-divisibility. In the second regime, all prime divisors of n are constrained to a highly restricted exceptional schema, where congruence conditions modulo a fixed squarefree modulus $M\mid (a+1)$ and the squarefree nature of $p_i-1$ limit the set of admissible primes. The exceptional schema is shown to be finite by purely elementary means, while the valuation obstruction relies on analytic information about the distribution of prime divisors of $p_i-1$. This formulation cleanly separates the unconditional density-zero theorem from the conditional finiteness result, ensuring analytic transparency while preserving the full logical structure of the argument. The Bombieri–Vinogradov theorem provides partial control over the placement of prime divisors in $p_i-1$, while stronger equidistribution hypotheses are invoked only where squared prime divisors are required to force valuation mismatches. Together, these tools illustrate how analytic number theory and multiplicative structure can be combined to study divisibility phenomena across additive shifts.

The results further highlight a sharp contrast between positive and negative shifts. For $a\ge 2$, the divisibility condition $\lambda \left(n\right)\mid n+a$ is sparse and, conditionally, finite. For negative shifts, explicit infinite families arise from simple multiplicative constructions. The borderline case $a=1$ remains open and appears to be governed by the infinitude of Fermat primes, marking a natural boundary between finite and potentially infinite behavior. Overall, the methods developed here provide a general template for investigating shifted divisibility problems involving other multiplicative functions, and demonstrate how local valuation structure can decisively constrain global arithmetic