Martini Quartets
A Martini Quartet is a structured set of four primes
(p₁, p₂, p₃, p₄) revealing a symmetry between prime gaps and twin midpoints.
It follows from the Martini Glass Conjecture that when the midpoint between a prime pair
is divisible by a twin prime, new twin primes appear symmetrically about that midpoint.
Definition
For primes p₁ < p₂ with gap g = p₂ - p₁ and midpoint
m = (p₁ + p₂)/2, if a twin prime t divides m
and both m−1 and m+1 are prime, then
(p₁, p₂, m−1, m+1) form a Martini Quartet.
Large Quartets
Extended searches using gmpy2 and PFGW have verified quartets
up to the 1500-digit scale with no counterexamples. Selected examples follow including to the best of my knowledge is the largest published Martini Quartet where the twin midpoint has 5547 digits and the Martini pair each 5553 digits.
digits=1051: Martini Quartet Twin Midpoint at
1240490543632701179964056606427489652593058303244439226199971344482099116902813781727762653599114337086413314536781215095168863178402619644962728141190099927966895258935192272144188133038410712448958060008855961012604751055276721479145472894553540264044787356158437848353296755055417185828835682989197032078157486757525780224743457823294380492953698140025058043862295974402234316919141543954248703055703973503245843799533790608983399129311012570716851060632521346796882185603102119795773555243288912569680383861510476076129896761038597104823334395746579707368482225523728785974089031983181632258748043754197902064478450715517756265177753202332617596074039614614320355434293698735898194209645056083915258013826168421161637867107217551825057959035462646117035068425573906266830470408412584284804693969484496861122556767454853285420222438337103196911577138183147333984910831672148562839043234592953978133152649832089912214042445548544333682171938881524776401625410854693880759134208314283740546391741238881701262926330134835365396480000000000000000000000
(primes at ±1) multiplier k = 8256, prime gap = 297914
digits=1207: Martini Quartet Twin Midpoint at
63993710294001276805557529589154809843215226985612248521669473098106937900561235407210807477540963184914278850712051708064939807282994807735412419520869958684792621949150595512759859742134635078077630802922098585286799363277479322489301598718521488726034072610907954479795533167138669569559467294369078179545098997326384131385045835664174153481440194018507601207023195323630415751276212537374876953359735886600418695582537913685962149964734155439470910759578077100985482499921144767302065725529502071406593000404226706480669026513263490972484385463691639307937545946989965534129771568318734682783812813432173355487328091991321634482186123562849731837131533728060739485483063901636502633185776597853257686959911042500429602267916199310633841172996887703269637903408322042849298919630832324733345957163663180320309626546546159308877913888715629706174672025161638542594183728706327946687174382837147481325522769321243054823906676239604443019930666192489109130980352856224116211641424151538898460734751691006723343522977197012824378746176789667260793353063907701059403674947526268431058627593426374563227028670412437667686699201586007711441006292309676741187580492283395270089742280845199356723200000000000
(primes at ±1) multiplier k = 21099, prime gap = 2352058
digits=5547: Martini Quartet Twin Midpoint at
320904337809065281078338897460401317444886590963578924278384732090008294808109640948884445675474177672492570762744158207242660153291959098506785306316592333933472396674182750078774582160199765762415045055924909360631504971956561314495597567745390655233523406173915842342324972908475587879685365224692302277613976022278684920228274629190092335163595652251587212638226680931950384659511315749114321173035528577769870439712501526343955007930950924957825496205997126923715363756426190475533038959811524715673617896609365628190804086228501715441696376299074133131997442527037905323444550150397760777665099707104633446454375909815313248506513970495439161724706962177428804145816855069053782503650478964998609394474187900056535973624332302687631529107742097286372913950646637291066151882382069592150808772305802477655073189700715626743584670051009777477311123842766122071443214183015573534930388696700424535995847539321463458780396216432923641274102262574981028654646207806897127759394596811425962275979541838722371704260983919144406701798009237699151282868261747697863574643216344799086205775028035872310469353402041102859567762404094050134529962941341730488637135550825657549292230566887827648176600786526657311789715447937409566331785123338476407278488005952316754510153949059628220839213026829068000764663729390000348255892092414821184964458571983763720733191192292824253627294810158609769072256333553085141439491528280297983600554669600851071642848755172537568697941977412935116147353309827681418564072299651162178283918381678758429405601569396977131232092714291863941054256218520503223724791994218771745905532554049888315818605617647993622763021941226197326976728489708409949525919334116831354008588092391041767422412731961767266467113649390789801075566888706039637635391583016357267440241131864882728393767669146601110260884499508555707147319700646673279347869378677358446068820623333563894803630496742056786097429735692723310931554147930717688677764398925209869640885915563833362591813171648594155757083329493960728673925323875117360397341233526100318183609657197893598638976202994601643672888095965906848650243401510159198496902194099536841973406741655969586292453900027465392559077084975800601545894207517665218623909195803987507760940803367204433835331488330318424579460927800198641852806549578505938606019461356771692506138690687200693659883356274078984746062578381435720118370988678184519955796615404815225172372696018170989435444961626486654092172588897609738345109273402637975051191219800114621296045799757212670228820656653091331544008379462963307278492576539747337439599310602263031345337394452574110870091369885344088863421032435695717307870908500610437844112302959690298546895465345204993919652310203818355022534741703982269591436753280093335163237449606921024251026370564969317153500884302554357371833310175796797562274377554690617822889081456048392178279468531633661314213639498426766640865232590978496789204285247938408710954023483657283521120505411645746145805591752241204358585812003518361414248980118598396462631999078210233077206008879115434513441765983699514786696714630372436219003100202389418981307679207088318585389553460891342184812600417897889664489807655204582343785819502395241197924641883413403893659461010246585051993569555452674716188673223865603708651659780491257097311924695061081625987598615913237379247936972579736572226024524430717980802660841086917913412550183775218235791890840530239448808569968937171991244366132983763765468150819574050771811808012545684218725798478322616039310885451904802426274646143836779481461713753559665822300977258165399347739434100904635250660423153804623690978998672212862321863287595209327487104909330969133138057562569437347214882374531562720437528040673301714725084020776107709876031589301467392766333507592489176546495651709681962855535017566644572777203396573159217584491327799076285550496884757540412440556589985158306956552050372030331778811181398272297438752018429843105756803775346166548034315492279463180408159167062424015456169586390666840938283262675924075743776262320746596343186321363984894522777428986834216664157961482098031473449539471220044967220284111653032007321535070715082559051734608709395939401522683650706214306391923367449839575118298617036758332480782010163040676664679302786839393246236276830680045033541463278818013222330207798289757664803801138839320111697431660293251711983664853432240325124561336027850941076592565406674412166364652948669402299604396534840972182875782973214909231711603112875289985018491105306881303261586908683018833143110717649396172108918372987432068357284007647439942318859578592207924336495641873685236539794341575229301443687670931213049386743511458734801083310687781860741648629381423743653565533882021834527689872874451721875198997205193019276859005242847067027336817107935068077803065143437275607720524039423127956436249545973362694708158916872891422188781137692000543470696557035251800101170826819458669703806631000413607842003962777031605576998164499861403169457051358983152309063316038842414888926566802142895764991526772437882684889327105105821506713190164566440168989212514813443702376976068401774586645783175483445791117437663826382489840237062665908677045546802159634065575416756538884648761157406304629820397474529262288278649587097778798285391103121041744133658188455158956873035322408162675225095260050021099484732147966199249869403082813422641783302083782826417664304111905595074667327745950779539718144000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
(primes at ±1) multiplier k = 652598, prime gap = 5851414
Verification
Quartets were verified via is_prime() tests (gmpy2) and
PFGW -q() PRP proofs.
The 1207-digit example above (K = 21099, Δ = 2352058) is recorded on the PrimePages, the standard registry of prime-number records: Prime Curios! 63993…00000, credited to Maughan.
Reproduce these results yourself: verify_quartet.py reconstructs a quartet from MT, K and Δ, screens it with gmpy2, then runs PFGW base-2 and base-3 PRP tests; scan_quartet.py finds the K or Δ that completes a quartet for a given twin midpoint. Both need gmpy2; the verifier also needs a PFGW binary on the path.
Open Questions
- Do specific K-factors constrain valid quartets?
- How does quartet density scale with digit length?