For every prime \(p\) satisfying \[ 10^6\leq p\leq10^{11}, \] the smaller distance between \(p^3\) and the square of either adjacent prime around \(p^{3/2}\) is at least 49,600. Equality occurs at \[ p=1587809,\qquad q_-(p)=2000771023, \] where \[ p^3=4003084686476516129, \] \[ q_-(p)^2=4003084686476466529, \] and \[ p^3-q_-(p)^2=49600. \]

The segmented sieve enumerated 4,117,976,315 prime bases in the stated interval. For each base it computed the exact integer square root of \(p^3\). Only four nearest-integer square gaps were at most 49,600. The displayed pair supplies the sole prime square among them. The other three square bases are even: \[ 2113144738,\quad59120053422,\quad1049747744368. \] Their gaps are 24,315, 17,767, and 25,895. Since their square bases are composite, they cannot enter the prime-restricted minimum.

Put \[ r=\lfloor\sqrt{p^3}\rfloor. \] The squares increase strictly with their nonnegative integer bases. Hence every \(q\leq r\) satisfies \(q^2\leq r^2\), while every \(q\geq r+1\) satisfies \(q^2\geq(r+1)^2\). It follows that \[ \min_{q\in\mathbf Z_{\geq0}}|p^3-q^2| =\min\{p^3-r^2,(r+1)^2-p^3\}. \]

Consequently, a prime square whose gap is at most 49,600 must first appear among these same two integer squares. The sweep can rule out a better prime square by classifying only the nearest-integer candidates whose raw gaps meet that threshold. Any adjacent prime farther away has a larger square gap on its side.

The factorizations \[ 1587809-1=2^5\cdot29^2\cdot59 \] and \[ 2000771023-1=2\cdot3\cdot19^2\cdot337\cdot2741 \] are complete. Trial division certifies the seven prime factors. For every distinct prime divisor \(s\) of \(n-1\), the replay gives a witness \(a\) satisfying \[ a^{n-1}\equiv1\pmod n, \qquad \gcd(a^{(n-1)/s}-1,n)=1. \] Since the factored part is all of \(n-1\), Pocklington's criterion proves each \(n\) prime.

Exact integer square root gives \[ \lfloor\sqrt{1587809^3}\rfloor=2000771023. \] The certified prime \(q\) is therefore the largest integer below \(p^{3/2}\), so it is automatically the adjacent lower prime.

For the complete candidate interval, both \(p\) and each adjacent prime \(q\) are odd. Their cube and square are odd, so every positive difference is an even integer. The incumbent therefore gives the rigorous global enclosure \[ 2\leq\min\leq49600. \] The exhaustive computation in this entry proves equality with the upper endpoint when \(p\leq10^{11}\). It makes no claim for \[ 10^{11}<p\leq10^{12}. \]

Hall's 1971 paper introduced the general search for small nonzero \(|x^3-y^2|\). Elkies developed a lattice-reduction method for rational points near curves, and Aanderaa, Kristiansen, and Ruud later gave an algorithm that detects every Hall-good example inside its stated search space. Those works allow arbitrary integer square and cube bases. The present problem imposes primality on both bases and asks for an exact finite minimum.

Baker, Harman, and Pintz prove that sufficiently large \(x\) has a prime in an interval of length \(x^{0.525}\). This gives asymptotic control of adjacent primes. Its scale is far larger than the interval in \(q\) corresponding to a square gap of 49,600 near \(p^{3/2}\), and the theorem does not identify the exact finite minimum here.

Focused searches for the pair \((1587809,2000771023)\), its exact cube, prime-restricted Hall problems, and nearest prime squares found no primary source reporting this candidate. The status and novelty of the full trillion-base problem remain unverified.

The program sieves odd base primes in blocks of \(2^{20}\) candidates. An initial floating-point square root of the 128-bit cube is corrected in exact arithmetic until it equals \(\lfloor\sqrt{p^3}\rfloor\). Thus floating-point rounding cannot alter a recorded gap.

For every base prime, the little-endian triple \((p,p^3-r^2,(r+1)^2-p^3)\) enters a streaming SHA-256 calculation. The digest is `89310fab45ad76bd3160d6ff906d25cbd12ffe40beb732c2f4df11ceb6a5d883`. The final prime is 99,999,999,977. Exactly four raw gaps are at most 49,600, and only the incumbent candidate passes the primality filter. The deterministic six-line output has SHA-256 digest `238ffc57522d272dd44c4b012be3256ba51c304ccd222d013360cd98010d9c69`.

The certificate uses a separate witness for each distinct prime divisor of \(n-1\). It verifies all small factor primes by trial division, then checks the Pocklington congruence and gcd conditions. The compact canonical report has SHA-256 digest `91d204497b47598411a8093cef855cd78b94d1725eae61cfb2f1b39ede71f35a`.