Which algorithms offer a credible local search for 114?

The strongest practical recommendation is to retain the exact norm-generator search, improve the amount of distinct, explicitly defined root space it checks per second, and keep Booker–Sutherland as the reference for coverage and comparison. The norm method makes selected large divisors accessible on a Mac. There is no published or locally demonstrated reason to expect it to find 114 within hours. Epoch's current open-problem page still identifies 114 as the smallest unresolved case. Epoch AI

Three objectives need separate evaluation: efficiently finding a candidate, completely excluding a bounded region, and predicting the time to a solution. A method can be excellent at the first and offer little evidence for the third. The present campaign is in that position.

The common exact problem is already one-dimensional after a divisor and root are selected. Put $s=x+y=\sigma D$, with $D>0$ and $\sigma\in\{-1,1\}$, and let $v=x-y$. Then

$$3sv^2=4(114-z^3)-s^3,\qquad z^3\equiv114\pmod D.$$

Given a root $r\pmod D$, substitute $z=r+qD$. A valid point requires an exact square and the parity condition needed to recover $x=(s+v)/2$, $y=(s-v)/2$. Root-first and norm-first approaches supply different collections of $(D,r)$ to this same arithmetic problem. Fast rejection of impossible $q$ values is useful regardless of which generator supplied the root.

Booker–Sutherland generates admissible divisors from prime powers, caches roots, combines them using implicit CRT enumeration, and uses local, reciprocity, and auxiliary-prime filters. Its ratio parameter is optimized against measured cost. The paper reports the 2019 search through minimum coordinate $10^{17}$, then a 2020 search with $z_{\max}=10^{19}$, $D_{\max}=10^{19}/54$. A separate completed-shard manifest for 114 is unavailable. Reciprocity alone retains about 0.962 of locally permitted residues for 114, so that particular enhancement offers only a modest further reduction here. Booker–Sutherland, §§2–5

The authors' implementation accepts a largest-prime-divisor interval and parallel thread count, supporting disjoint factor-based work packets. It requires GMP and primesieve. These are concrete advantages for an exhaustive reference implementation; it is already sophisticated numerical software. Reference implementation and invocation contract

The recorded single-core sandbox calibration, $4.6\times10^{-8}$ core-seconds per unit of $D_{\max}$, extrapolates to approximately 270 core-years for $D_{\max}=10^{19}/54$. This extrapolation is not a Mac benchmark. Dividing by 12 gives about 22.5 calendar years under ideal scaling and equal core performance. It does not establish the cost of the selective norm search, and redoing that whole historical rectangle would add little discovery value.

The norm approach changes which divisors are affordable, not the final equation. Grantham–Walsh constructs a progression from a cubic-field norm and converts its remaining search into integral points on an elliptic curve. Its worked large $k=3$ example has generator coefficients only in the thousands but progression index 4,360,888. The preprint gives conflicting estimates for that example: 137 core-years in its introduction and approximately 11 core-years later, using about 3 milliseconds per curve and roughly $10^{11}$ curves. Neither is a blind 114 benchmark. The paper explicitly reports a substantial unsuccessful effort on 114. Grantham–Walsh, pp.2–4

For the specialization to 114, $\alpha^3=114$ and

$$N(a+b\alpha+c\alpha^2)=a^3+114b^3+12996c^3-342abc.$$

When the required inverse exists, the adjugate coefficients give a cube root modulo $D=|N|/\ell$, with $\ell\in\{1,5,25\}$. The certified class-group calculation proves that these choices can principalize every root-ideal class. This is support for every ideal class, not coverage of every root ideal: the finite coefficient boxes and offset planes still contain only selected generators. Arbitrary sampled generators also require the stated inverse check. Local field and unit analysis

The completed initial phase2 run checked 169,212,216,041 curve-intervals in 23.425 aggregate CPU-hours. That establishes real local throughput for its particular distribution. It does not mean 169 billion statistically interchangeable opportunities for a solution, or 169 billion different divisors. Different disjoint ratio bands may revisit the same root. More than eleven billion coefficient positions also encountered a noninvertible root coefficient; these remain a separately reported limitation, not a proof that the corresponding divisors have no solutions. Phase2 report, phase3 safety review

There is an important nuance in the lattice literature. Elkies' basic near-curve algorithm has heuristic cost $(N+\delta N^3)\operatorname{polylog}N$; its output-optimal regime requires $\delta\gtrsim N^{-2}$. For cubes, that corresponds to a residual window of order $N$, much wider than fixed 114. However, §§2.3 and 3.2 also propose a Veronese embedding with rank-six lattices: a heuristic $N^{12/13}\operatorname{polylog}N$ search for residuals of order $N^{3/13}$. Elkies says this refinement was not implemented and highlights its potentially substantial constants. Elkies, §§2.3 and 3.2

Thus it would be incorrect to dismiss every lattice variant by citing only the basic wide-residual condition. Nevertheless, substituting $N=10^{17}$ and $10^{19}$ into the improved exponent gives scales of approximately $4.9\times10^{15}$ and $3.5\times10^{17}$, before the lattice and logarithmic costs. The exponent-only gains over linear work are about 20 and 29. These are scale illustrations, not rigorous lower bounds or runtime predictions. They offer no credible hours-scale full search. Moreover, the coordinate bounded by the geometric algorithm differs from the minimum-coordinate bound central to the divisor method.

The geometric method has genuine practical achievements. Huisman used the Elsenhans–Jahnel implementation of Elkies' approach, covering the Fermat curve with narrow parallelograms and reducing lattices to enumerate the associated integer points. His 2016 campaign reached the $10^{15}$ height scale for targets below 1000, found 966 new representations including the first for 74, and reports slightly more than $10^5$ computation-hours. This supports its value for collecting many targets together; it supplies no measured advantage for one unresolved target at a much higher frontier. Huisman, §§2–3 and p.1 computation note

A bounded elliptic search must not be mistaken for eliminating a whole curve. PARI's hyperellratpoints accepts a specified height or interval and searches within it. An empty result does not certify absence at all heights. Complete integral-point methods need additional arithmetic certification and can fail to finish cheaply. Booker–Sutherland report all-height divisor exclusions only through $D\le100$ unconditionally and $D\le20,000$ under GRH for the unresolved cases. PARI documentation, Booker–Sutherland, Remark 3.8

A useful research branch would attempt a small, preregistered collection of cheap certified curve exclusions and measure the actual finite or infinite region removed per CPU-second. Running a general curve solver on every frontier root is not justified by the literature. Its setup cost could overwhelm the native sieve, and the usual rank or root-number statistics do not by themselves predict an integral point in the required progression and height range.

Recent work does not establish a superior search algorithm for 114. Browning–Wilsch develop density heuristics for cubic surfaces and compare with numerical data. Their discussion supports the logarithmic-growth perspective but does not calibrate the fertility of a selected norm-generator shape. Averaging across targets or an entire height region does not prove equidistribution inside the campaign's three chosen offset families. Browning–Wilsch, especially §§1,5,10; published 2025

The December2025 seed-equation preprint first assumes a known representation and studies lines through it. In its §6 treatment of missing seeds, it instead starts from another integer and searches a cubic polynomial for suitable integer roots; it gives no complexity improvement or new 114 solution. Algebraically, substitution along a line preserves the need for integrality. The reduction in polynomial degree depends on already having a point on the desired surface. This is a useful identity mechanism for applicable families, not an established shortcut to the first point for 114. N. K. Wadhawan and P. Wadhawan, §§2,6

Recent results for three positive cubes and for polynomials over finite fields answer different counting questions. They do not provide a finite short-time procedure for a specific integer such as 114 with enormous cancelling signed coordinates. Maynard, 2026, Browning–Glas–Wang, function-field results

A defensible next strategy has three components.

1. Continue producing exact, distinct $(D,r,q\text{-interval})$ coverage with the norm backend and proven local cuts. Promote optimizations only after identity-equivalent tests and a matched benchmark on the same domain. Keep unresolved inverse cases and any unsupported domain outside certified exclusions.
2. Make coverage geometry more systematic. Future domains should be blocked by normalized unit phase $\log(|\sigma_1\gamma|/|N\gamma|^{1/3})\pmod{\log\beta}$, with explicit norm and ratio bounds. This gives an interpretable diversity axis and principled deduplication. Equal phase width is not equal root mass or equal solution probability; finite coefficient boundaries still matter. Existing derivation and scope
3. Maintain an independent root-first comparison on precisely defined, feasible factor packets, including packets in the intended large-divisor regime. Compare accepted root coverage, ratio coverage, CPU time, and overlap with the norm domains. Small-divisor timing alone is insufficient. This can reveal whether a new family improves access to unvisited root space rather than simply increasing an internal counter.

The most meaningful possible breakthrough would be a theorem or independently validated structural bias that identifies solution-bearing roots much more efficiently. The strongest immediately testable engineering improvement is cheaper exact elimination before expensive root and square work. These are different claims and should be reported separately.

Neither “70 years” nor “11%” is presently a supported answer forecast. A time estimate for exhausting a finite coefficient domain says when that domain finishes; the domain may contain zero solutions. No observed 114 hit currently calibrates its fertility. Under an explicitly assumed Poisson model with the published asymptotic constant $\rho\approx0.0584593$, a complete, similarly shaped divisor expansion by a factor $f$ would give $1-e^{-\rho\log f}$. An eightfold expansion gives approximately 11.45%, reasonably rounded to 11%; the limitation is its application to the sparse norm sample, which does not complete that region. Doubling gives approximately 4%, and a decade approximately 12.6%, under the same conditional model. Density constant and search-shape model: Booker–Sutherland, §§2,4

Current evidence supports credible exact local research, useful new coverage, and uncalibrated discovery odds. There is no proof that the process will terminate with a solution, no defensible hours-scale promise, and no evidence that a failure in one arithmetic neighborhood makes a nearby neighborhood fertile. The algorithm can learn cost and coverage efficiently while mathematical discoveries remain uncertain.