Write \(x_0=0\) and \(x_{k+1}=x_k^2+1\pmod p\). An exhaustive increasing-prime sweep found that \[ p=1{,}229{,}237 \] is the first prime whose orbit contains more than 5000 distinct residues. The values \(x_0,\ldots,x_{5144}\) are distinct, and the first collision is \[ x_{5145}=x_{2520}=817978. \] Thus the preperiod is 2520, the period is 2625, and their sum is 5145. The 94,997 smaller primes all have total orbit length at most 5000. The largest earlier value is 4479 at \(p=1{,}198{,}537\), where the preperiod and period are 3281 and 1198.

Primality has two independent checks. Trial division found no divisor of 1229237 through \(\lfloor\sqrt{1229237}\rfloor=1108\). A Pocklington certificate uses \[ p-1=2^2\cdot17\cdot18077. \] The numbers 17 and 18077 are prime by trial division through 4 and 134. With base 2, one has \(2^{p-1}\equiv1\pmod p\). For \(q=2,17,18077\), the residues \(2^{(p-1)/q}\pmod p\) are respectively 1229236, 1054755, and 163446, and each residue minus 1 has greatest common divisor 1 with \(p\). This is a complete Pocklington certificate.

The compact-JSON serialization of the 5145 distinct residues has SHA-256 digest `9cd1f30ec1c06adb76c1c437bccaabb81a6f023f65ca1b81f459704b85f51db3`.

Heath-Brown studies the directed functional graph of a quadratic polynomial over \(\mathbb F_q\), separating each trajectory into its preperiodic tail and cycle. Corollary 1 gives an unconditional \(O(q/\log\log q)\) bound for the first recurrence of the critical orbit. Corollary 2 gives the corresponding cycle and preperiod bounds for reductions of fixed positive definite quadratics, including \(X^2+1\).

This theorem supplies asymptotic context for the finite search. The paper contains no finite record table identified during the focused search, so the witness and minimality claim above rest on the reproduced computation.

The C program sieves every prime through 1229237, visits them in increasing order, and iterates \(x\mapsto x^2+1\pmod p\). Epoch-stamped arrays retain the first index of each residue without clearing the arrays between primes. A repeated residue gives the total distinct count, its first index gives the preperiod, and their difference gives the period.

The program returned `tested=94998 witness=1229237 total=5145 mu=2520 lambda=2625 repeat=817978 previous_best=4479 previous_p=1198537 previous_mu=3281 previous_lambda=1198`. An earlier run over the same range also reproduced the candidate datum \(p=192553\), total 1808, preperiod 293, and period 1515.

The verifier constructs the orbit with a Python dictionary and obtains total 5145, preperiod 2520, period 2625, and repeated residue 817978. Floyd's cycle algorithm uses constant auxiliary space and independently returns the same preperiod, period, and cycle entry. The last new residue is 1039286, which maps to 817978.

A direct divisibility test through 1108 certifies primality independently of the sieve. The verifier also checks the displayed Pocklington congruences and the orbit digest. Its first ten residues are 0, 1, 2, 5, 26, 677, 458330, 848734, 540676, and 769059.