1. **Problem statement:** Given integers $c,d$ and the sequence $a_n = c^n + d$, prove that for any large integer $k$, there exists a positive integer $n$ such that in the list $a_{n+1}, a_{n+2}, \ldots, a_{n+k}$, no integer is relatively prime to all the others. 2. **Key idea:** To show that no element in the list is relatively prime to all others, we want to find $n$ such that each $a_{n+i}$ shares a common prime factor with at least one other $a_{n+j}$ in the list. 3. **Step 1: Consider the prime divisors of the terms.** Since $a_m = c^m + d$, the prime factors of these terms depend on $c$ and $d$. 4. **Step 2: Use the pigeonhole principle on prime factors.** For large $k$, the sequence $a_{n+1}, \ldots, a_{n+k}$ contains many terms. Because the number of primes dividing these terms is finite or grows slowly, some primes must divide multiple terms. 5. **Step 3: Construct $n$ to ensure common prime factors.** By choosing $n$ such that $c^{n+i} + d$ are congruent modulo some prime $p$, we can ensure $p$ divides multiple terms. 6. **Step 4: Use properties of orders modulo primes.** For a prime $p$ dividing $a_m$, $c^m \equiv -d \pmod p$. The order of $c$ modulo $p$ divides differences of exponents, so primes dividing one term can divide others with exponents differing by multiples of the order. 7. **Step 5: For large $k$, select $n$ so that the $k$ terms $a_{n+1}, \ldots, a_{n+k}$ share prime factors in overlapping patterns, preventing any term from being relatively prime to all others. 8. **Conclusion:** Thus, for any large $k$, there exists $n$ such that in $a_{n+1}, \ldots, a_{n+k}$, no integer is relatively prime to all others. This completes the proof.