Recursive sequence $a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$ generating twin primes

I have been analyzing a recursive sequence based on the greatest common divisor that acts as a dynamic sieve for twin primes. It shares structural similarities with Rowland's prime-generating sequence but targets the difference of squares.

For any integer $n \ge 2$, define the sequence $\{a_k\}_{k=1}^{\infty}$ with the start value $a_1 = n^2 - 1$ and the recurrence relation:

$$a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$$

Because $a_k$ strictly decreases whenever $a_k > 0$ and drops by integer amounts, the sequence must reach $0$ in finite steps. Let $T(n)$ denote the termination time, defined as the smallest index $m$ such that $a_m = 0$.

Proven Properties

Two fundamental properties of this sequence can be established:

1. Boundedness: $T(n) \le n^2$. This follows trivially since $\gcd(a_k, \dots) \ge 1$, meaning $a_k$ decreases by at least $1$ at each step from $n^2 - 1$.
2. Parity: $T(n) \equiv n \pmod 2$. By tracking the parity states, one can show that $a_k \equiv n+k \pmod 2$ for all $k$. At termination, $a_{T(n)} = 0 \equiv n + T(n) \pmod 2$.

The Conjecture

Define the terminal offset $P(n) = n + T(n)$. Based on empirical testing, I have formulated the following conjecture:

Conjecture: For all $n \ge 2$, $P(n) - 1$ and $P(n) + 1$ are twin primes.

Structural Analysis and The Descent Invariant

To understand the sieving mechanism, we can define a tracking variable representing the candidate twin prime center:

$$I_k = a_k + n + k$$

Let $g_k = \gcd(a_k, (n+k)^2 - 1)$. Evaluating $I_{k+1}$ reveals a descent invariant:

$$I_{k+1} = a_{k+1} + n + (k + 1) = (a_k - g_k) + n + k + 1 = I_k - (g_k - 1)$$

This shows that $I_k$ is monotonically decreasing.

• If $g_k = 1$, then $I_{k+1} = I_k$. The candidate center remains locked.
• If $g_k = d > 1$, then $I_k$ drops by $d - 1$.

We can determine exactly what triggers a drop by rewriting the GCD argument. Substitute $n+k = I_k - a_k$ into the second argument of the GCD:

$$g_k = \gcd(a_k, (I_k - a_k)^2 - 1) = \gcd(a_k, I_k^2 - 2I_k a_k + a_k^2 - 1)$$

By the Euclidean property $\gcd(A, B) = \gcd(A, B - cA)$, we can eliminate all multiples of $a_k$:

$$g_k = \gcd(a_k, I_k^2 - 1)$$

The sequence tests a candidate twin prime center $I_k$ against potential divisors up to $a_k$. If it finds a common divisor $d > 1$ with $I_k^2 - 1$, it rejects $I_k$ and jumps down to a new candidate.

The Obstruction: The Final Glide

Let $j$ be the largest index such that $g_j > 1$. For all $k > j$, $g_k = 1$. Let $L = a_{j+1}$ denote the value of $a_k$ at the start of this final, uninterrupted descent to $0$.

During this "final glide," $I_k$ remains perfectly constant at its final value, $I_{last} = P(n)$. As $a_k$ drops from $L$ down to $1$ in increments of exactly $1$, the sequence verifies that:

$$\gcd(x, P(n)^2 - 1) = 1 \quad \text{for all } x \in \{1, 2, \dots, L\}$$

This guarantees that $P(n)^2 - 1 = (P(n)-1)(P(n)+1)$ has no prime factors $\le L$.

However, for $P(n) \pm 1$ to be mathematically guaranteed as prime, the residual sieve limit $L$ must strictly satisfy:

$$L \ge \sqrt{P(n) + 1}$$

While the slow, logarithmic decay of $a_k$ compared to the linear growth of $n+k$ empirically preserves this bound, I have found no algebraic or analytic method to prove that the sequence never "prematurely terminates".

Questions

1. Is this specific class of GCD sieve for polynomial differences known or studied in the literature?
2. Are there known number-theoretic mechanisms, sieve methods, or probabilistic bounds capable of proving that the residual sieve limit always satisfies $L \ge \sqrt{P(n) + 1}$, or is the existence of false positive semiprimes analytically inevitable without assuming strong bounds on prime gaps?