math/big: ProbablyPrime panics when the Lucas parameter search exceeds its bound #81487

The input is the 4,302-digit integer

$$
M = 8 \prod_{\substack{q \le 10002 \ q\text{ odd prime}}} q, \qquad N = 655M + 1.
$$

In src/math/big/prime.go, ProbablyPrime runs Miller–Rabin before probablyPrimeLucas. This $N$ passes the Miller–Rabin gate for ProbablyPrime(0), ProbablyPrime(1), and ProbablyPrime(20), but all three calls panic in the Lucas parameter search.

The search tries discriminants

$$
D = P^2 - 4 = (P - 2)(P + 2), \qquad 3 \le P \le 10000,
$$

looking for a Jacobi symbol of $-1$. Every prime factor of such a $D$ is at most $10002$. By construction,

$$
N \equiv 1 \pmod{8}, \qquad N \equiv 1 \pmod{q}
\quad\text{for every odd prime }q \le 10002.
$$

For each of these odd primes, quadratic reciprocity gives

$$
\left(\frac{q}{N}\right) = \left(\frac{N}{q}\right) = 1,
$$

since $N \equiv 1 \pmod{4}$. The supplementary law also gives $\left(\frac{2}{N}\right) = 1$. Multiplicativity therefore yields

$$
\left(\frac{P^2 - 4}{N}\right) = 1
\quad\text{for every }3 \le P \le 10000.
$$

Neither the j == -1 success branch nor the j == 0 factor branch is taken. The loop reaches p > 10000 and panics, printing the full input. A regression test can construct $N$ as above rather than embed its decimal expansion.