Show HN: Using GPUs for other bs than AI – important prime record for the geeks

A developer released BEACON, an open-source GPU-accelerated toolkit that hunts record prime gaps by centering searches on primorial-based numbers of the form n₀ = k·zd#/d rather than scanning random ranges, and prices each record's cost in advance. The kit, published on GitHub under the handle djuk8, runs on any Linux machine with NVIDIA GPUs and the CUDA 12 toolkit; a healthy RTX 5090 reads about 12,000-14,000 windows/s, and since version 8.57 a fresh box hunts one residue class of k mod 2310 from k = 10^16 + seed·10^15. Found gaps are re-derived by tools/notary_v2.py, certified with ECPP by tools/certify_batch.py, and submitted to the Prime Gap List, which tracks over 120,000 first-known and record gaps scored by merit (gap / ln(p₁)).

Any Linux machine with NVIDIA GPUs, the CUDA 12 toolkit, gcc, git, python3 and pari-gp step by step, including one-GPU machines and how to submit what you find: SETUP.md https://github.com/djuk8/beacon/blob/main/SETUP.md : python3 -m pip install numpy gmpy2 git clone https://github.com/djuk8/beacon ~/lab/beacon && cd ~/lab/beacon bash deploy/fresh box.sh --seed N N in 1..999, your own k-range see SETUP.md fresh box.sh extracts the kit, clones NVlabs' CGBN and builds the GPU Fermat driver for your card, writes the venue and seed files, builds the residue cache about 15 minutes , builds the C and CUDA sieve, runs the identity gate the GPU sieve against an independent CPU sieve, byte for byte and launches the hunt with a watchdog that relaunches anything that dies. Twenty minutes later: bash ~/beacon/p3/o1 kit/daily.sh prints throughput per four-hour slice, the fossil audit, the harvest and the day's best gaps. A healthy box reads about 12,000-14,000 windows/s per RTX 5090 with all GPUs at 100% the first hour ramps . Every gap the hourly harvest walks lands in ~/beacon/raid610/campaign ledger.log ; tools/notary v2.py re-derives any gap from its tuple, tools/certify batch.py proves the endpoints with ECPP, and the lines those tools print go to the Prime Gap List's submission server. The seed picks the k-range: since v8.57 a fresh box hunts one residue class of k mod 2310 317, and 1993 on a second sieve card from k = 10^16 + seed·10^15, with rings 1 and 2 sieved on the GPU; machines must not share a seed. --mixed gives the older all-classes mode k0 = 10^12 + 8·10^9·seed . The current tree is stamped by STAMP 20261104.sha256 ; sha256sum -c STAMP 20261104.sha256 --quiet verifies it. The knob files behind it SIEVESTEP / SIEVECLASS / SIEVEKBASE / SIEVEBID, SIEVERINGS are described in KIT v8.56 NOTE.md and KIT v8.57 NOTE.md . A prime gap is the distance between two consecutive prime numbers. Between 23 and 29 there is a gap of 6. Gaps grow — on average — like the natural logarithm of the numbers involved, so raw gap size mostly tells you where you looked, not how remarkable the find is. The community therefore scores gaps by merit : merit = gap / ln p₁ — how many times larger the gap is than an average gap at that height p₁ is the prime that starts the gap . A merit of 20 means "twenty average gaps' worth of nothing, in one stretch." The prime gap list https://github.com/primegap-list-project/prime-gap-list tracks every first-known and record gap — over 120,000 entries — and several leaderboards sit on top of it: the all-time merit table highest merits ever, any size , size tables at merit thresholds "largest gap with merit ≥ 20 / ≥ 30" , and per-gapsize first-occurrence records smallest starting prime known for each gap size . People have hunted these tables for decades, mostly by searching enormous ranges and hoping. BEACON's thesis: hope is the expensive part. Record gaps can be designed — placed where mathematics makes them cheap — and the cost of any record on any table can be priced in advance by a measured law. This repo is the law, the machinery, the verification chain, and the receipts. a Choose where the gap lives. Instead of scanning random territory, center the search at numbers of the form n₀ = k · zd / d where zd is a primorial the product of all primes up to zd , d is a small divisor of it, and k is a small integer. Around such n₀, divisibility is not random: every prime p ≤ zd except those dividing d automatically divides n₀, so it can only "interrupt" the gap at positions j that are multiples of p. The neighborhood of n₀ is born mostly empty. The offsets j where a prime could live — the survivors — are few and known in advance. b Sieve, don't test. A cheap sieve all primes up to Q, typically 10⁵–10⁹ removes every position with a small factor. What remains gets a real primality test. At our working heights, 97–99% of positions never reach a test. c Test the survivors until the walls appear. Walk outward from n₀ in both directions; the first surviving position that turns out to be prime on each side is a fence . The distance between the left and right fences is the gap, and it is born verified : every interior position is either sieved-out provably composite or was tested and failed. d Price it before you run it. The number of independent attempts windows — distinct values of k needed before both fences land far enough out follows a measured exponential law. Knowing that law, plus the measured speed of the hardware, converts any target on any leaderboard into a number: so many GPU-days. Then a record hunt is a purchase decision, not a lottery ticket. The load-bearing results, each with its measurement provenance in docs/ and its code in this repo: - The fence-rate law Buchstab . The rate at which candidate fences turn out prime is λ u = e^γ · ω u per unit of merit, where ω is the Buchstab function and u = ln j /ln zd measures how far outside the "free" region you are. Confirmed three independent ways, including a 94,885-fence out-of-sample test with zero fitted parameters χ² p = 0.19 and a preregistered forward prediction confirmed at 6–7σ in both directions. The classical Mertens-style rate is the u→∞ limit and misfits by up to 21× at high merit. - The unified cost law. Put every leaderboard on one axis with u m, D = 1 + ln m/2 /ln D·ln10 m = merit, D = decimal digits ; then the exponential cost is E m, D = m·σ u with σ a two-parameter curve whose leading coefficient is a = 1.8169 ± 0.0367 — equal to e^γ within 1.0σ 26 calibration points, R² = 0.994; deriving this constant is the open theorem . Out-of-sample checks at three heights land within 0.3–2.6%. - The d-axis. Dividing the primorial by small d relocates which positions each small prime can kill. Drop-2 d = 2, k odd doubles the free desert without surrendering the floor and halves the near fence rate to e^γ·ω 2 — measured 1.82× more merit per test. The construction is folklore among record holders; the pricing and the identification of the rate as a Buchstab value are, to our knowledge, new. - Survivor structure at extreme height. At 26,599 digits with zd ≈ 61,000, every even survivor j in the working window must be 2q or 4q for a prime q zd — the candidates are forced almost-primes, and the survivor density 0.0167/unit, vs naive 0.0407 follows from that structure exactly. - The staircase. With the law calibrated, each +1 of merit costs ×2.4–2.6 in compute at fixed height, and the entire all-time merit table — from its entry bar to a fixed multiple past its nine-year crown — is priced in fleet-days by one configuration. - Dead ends, kept dead. Candidate scoring: null at 14σ. Wing- weighted covering: net zero at production. Deeper sieving beyond the regime boundary: cost, not benefit. Every retired multiplier died against a finished control, and stays in the record. One liturgy runs every campaign, at every height, on CPUs or GPUs: php design - make - drive - fold - ... rings 2..R ... - finish rings sieve PRP verdicts walk fences - Design theater/ /ladder .py , exchange/ . Choose zd, d, k digits, and a ladder of rings — nested windows sized so each costs about the same expected number of fence-primes E ≈ 1.7 per ring . A k that shows a prime in ring 1 is discarded immediately; survivors advance outward. The ladder concentrates testing on the ~e^-E fraction of ks that could possibly be records. - Make theater/raid610/scanhunt v0.py + tools/fastmake.py . The sieve factory: for a block of ~8,300 ks, compute every ring survivor. Vectorized per-prime across the whole block; byte-identical gated to the reference per-k path. - Drive. Fermat/PRP-test the candidates. On GPUs: cuda/g1 fermat v2.cu — a Montgomery-arithmetic kernel with a Gerbicz error check on every test zero alarms across 18.5M+ verdicts ; measured 516–523k tests/s/card at the 2048-bit class. On CPUs at extreme height: tools/pfgw drv.py shards candidates across PFGW processes. - Fold. Read the verdict file, count fence-primes per k, advance survivors to the next ring. Every file carries a magic number and a count; mismatches fail loudly. Fold statistics E per ring are the live health metric — the campaign certifies itself every block. - Finish / walk. A k that clears the full ladder is a finalist : walk its true fences and record the gap. tools/shard walk.py does the same for salvage — see below — and its sieve is proven correct by exact set-equivalence against an independent implementation. - Salvage tools/salvage v0.py , tools/shard walk.py . The campaign's exhaust is ore: a k killed at ring r still has a certified prime-free core out to ring r−1's edge, and often one fence far beyond. Walking discarded ks recovered, among others, a size-table crown and an all-time-merit entry — at seconds per k. Doctrine: sweeps touch only fossils age-gated manifests , never a live theater's working files. - Verification chain. 1 Sieved-out interior = provably composite. 2 Every tested candidate has a GEC-checked or PFGW-issued verdict. 3 Finds are independently re-proved separate library, both endpoints before banking. 4 The public record's notary re-proves endpoints and re-sieves the interior on independent hardware at submission. Nothing depends on trusting a single program. - September machinery RC11/RC12 . A persistent bucket sieve tools/sieve/ , FK lattice walk, one 24-byte record per prime and its GPU port cuda/gbs.cu sieve every window to 10¹⁰ — one card sieves for the whole box theater/raid610/gbsd.py , the sieve card , byte-identical to the CPU reference gates G-BSD, G-GBS, G-SV incl. stop-and-resume . A full-window sparsity predictor residue tables for primes to 10⁵ ranks the pool before testing: ×1.42 crown-λ per tested window gate G-PRED . A ladder manager ladder.py , one process per card replaced 700 shell loops, replay-gated identical G-LADDER . Feeders and drivers pinned to reserved cores. daily.sh is the one-screen production read; harvest hourly.sh walks every fossil and keeps the campaign ledger. Three "theaters" ran this architecture at three heights details, configs, and preregistrations in docs/ and theater/ : | theater | height | target board | outcome | |---|---|---|---| | raid610 | ~611 digits | all-time merit table | the record: 59,724 @ merit 42.5192 previous record 41.9388, 2017 ; 2: 58,416 @ 41.5882 ; 20 certified first-occurrence records in the list, ~25,000 further claims in the ledger | | strike24k | ~6,152 digits | largest gap @ merit ≥ 30 | crown : 426,520 @ 30.1156 +11 size records at merit 25–30 | | mono | ~26,599 digits | largest gap @ merit ≥ 20 | 1,383,648 @ 22.59 , 1,348,726 @ 22.02, 1,263,196 @ 20.62 | Plus the ≥2,500-digit merit crown 209,420 @ 33.638 and its guard 201,646 @ 32.385 from the 2,704-digit campaign. The record and its runner-up came out of the salvage walk over near-miss windows — asymmetric gaps that a symmetric ring ladder discards — a channel the cost law had not priced; the paper does. All entries were submitted under mvojvoda on 2026-09-22 and re-verified by the list's own server; endpoints of the certified batch are ECPP-proven records/ . Every find above was priced before it was hunted , and the realized costs sit inside the predicted bands — the cost ledger hardware, hours, dollars is part of the release. Throughput of record September 2026, one 4×RTX 5090 box : 44–45k windows/s sustained, ~64k crown-effective, with the CPU nearly idle. - Preregistration. Campaign configs, expected bands, and success criteria are written down dated before data collection; amendments are dated; verdicts PASS / REFUTED are filed strictly — including the refuted ones. - Gates. No component touches a real pool before passing a known-answer corpus, a bit-exact comparison against an independent implementation, and a fault-injection drill. Gate scripts live in gates/ ; the rule "a green gate on code the gate does not cover is worse than no gate" is enforced culturally and in review. - Lineage. LINEAGE.md maps every canonical tool to its source machine, SHA-256, and the gate that certified it. - Corrections ledger. Mistakes are filed by name — including the AI assistant's — with mechanism and date. Several appear in the papers because the pattern underpowered data reads as structure; controls must finish is itself a result. - Find files. Every find is reconstructible from one small text file k, zd, d, fences ; anyone with this repo can re-prove any claim end-to-end. tools/ canonical operational tools sieve factory, drivers, walkers, salvage, submission formatter, notary client gates/ known-answer + equivalence + fault-injection gates exchange/ the pricing instruments: pricer v1 cost law - GPU-days , merit price v0 bridge-exact certifier , fit sigma v1 the unified-law fit , audit + resonance tools theater/ per-campaign configs and loops raid610, mono, strike24k cuda/ GPU kernel sources g1 fermat family, gbs.cu GPU sieve + gate harness tools/sieve/ the persistent bucket sieve C : bsieve.c, bs api.c, fk walk.c + referees records/ what is on the Prime Gap List: submission lines, the treasury by venue, record endpoints, verification chain certificates ship with the data archive deploy/kit v8/ the current kit as deployed o1 kit v8.57; notes KIT v8.5x NOTE.md docs/ mission, preregistrations, cost calculus, novelty sweep, retrodictions, session handoffs history/ frozen snapshots of earlier eras arch2704 construction campaign; full deployed-tool archive — cited, not run LINEAGE.md the tool provenance table Not in this repo: the vault find files, ledgers, campaign states, raw manifests, databases ships as a separate data archive with a DOI. deploy/kit v8/ holds the current kit o1 kit v8.57; KIT v8.5x NOTE.md describe each step since v8.54 - the machinery that set the records; deploy/fresh box.sh stands it up quick start below . history/kit v1 2026-08/ is the August kit, kept as history. The field already had gap construction in the folklore, excellent sieves — notably S. Troisi's combined sieve arXiv:2012.03771 https://arxiv.org/abs/2012.03771 , which out-sieves ours and is cited with gratitude — numerical record- probability tooling, and the /d trick at the top of the record list. What we believe is new: the measured fence-rate law and its Buchstab identification, the unified cost law with its e^γ coefficient, the one-equation design grammar zd ≈ 1.01·ln n₀ , frontier audits of existing records on their own artifacts, and a fully verified, priced, reproducible pipeline. All novelty claims are phrased "to our knowledge"; the audit numbers stand as measurements regardless. | term | meaning | |---|---| | primorial, zd | product of all primes ≤ zd | | n₀ | the gap's center: k · zd /d | | k | the small free integer scanned; one k = one window | | d | small divisor of zd the "d-axis" design dial | | j | offset from n₀; candidates are n₀ + j | | survivor | a j with no factor ≤ Q the sieve bound | | fence | first prime found on each side; the gap's endpoints | | merit | gap / ln p₁ | | ring / ladder | nested windows; ks advance outward only while prime-free | | E | expected fence-primes over a region; cost ≈ e^E windows | | β | probability a survivor is prime ≈ e^γ·lnQ/ln n₀ | | fold | verdict-ingestion step; produces per-ring statistics | | finalist | a k that cleared the whole ladder | | salvage / crumb | walking discarded ks' certified cores for gaps | | window rate | windows/second; the campaign's speedometer | | GEC | Gerbicz error check — per-test hardware-fault detection | | born verified | interior composite-certified at find time | | claimcheck | querying the public list + staging queue before submitting | | theater | one campaign at one height raid610 / mono / strike24k | | bank / freeze / parade | finds are stored, not submitted, until one coordinated release | BEACON is M. Vojvoda's campaign records under the handle mvojvoda . Python 3.10+, numpy gmpy2 for verification . GPU path: CUDA 12+ kernel source in cuda/g1 fermat v3.cu; built by deploy/fresh box.sh . CPU-height path: PFGW sourceforge; expression syntax used throughout .