History of Bateman-Horn constants of monic quadratic polynomials

 Title: Bateman-Horn constants of monic quadratic polynomials-Definition: For a monic irreducible quadratic polynomial $f\in\mathbb Z[x]$ with no-  fixed prime divisor, this table stores the Bateman-Horn constant $C(f)$ CITE{WikiBH}-  CITE{BH1962} in the normalization of CITE{formula-product}.+Definition: For a monic irreducible quadratic polynomial $f\in\mathbb Z[x]$ whose+  values have no common prime divisor, the Bateman-Horn constant $C(f)$, also called+  the Hardy-Littlewood constant of $f$, is $C(f)=\prod_p\dfrac{1-N_f(p)/p}{1-1/p}$,+  where $N_f(p)$ is the number of roots of $f$ modulo $p$ CITE{BH1962} CITE{WikiBH}. Parameters:   f:

       Q$, and no prime divides every value $f(n)$ Comments:-  comment-normalization: The table stores $C(f)$ itself. Since every polynomial here-    has degree $2$, the Bateman-Horn asymptotic in CITE{formula-asymptotic} has the-    leading constant $C(f)/2$.-  comment-range: The table holds every polynomial $x^2+a$ with $-30\leq a\leq30$ and-    $-a$ not a square, and every polynomial $x^2+x+a$ with $1\leq a\leq41$ and $a$-    odd. The even values of $a$ in $x^2+x+a$ are omitted because all values of the-    polynomial are then even.-  comment-conjectural: The Euler product defining $C(f)$ converges for every polynomial-    in the parameter domain. The asymptotic interpretation as a density of prime values-    remains conjectural.+  comment-normalization: The entries are $C(f)$ itself. Since every polynomial here+    has degree $2$, the Bateman-Horn asymptotic in CITE{formula-asymptotic} has leading+    constant $C(f)/2$.+  comment-range: For even $a$, every value of $x^2+x+a$ is even, so those polynomials+    are excluded; for odd $a$, no value is even, $N_f(2)=0$, and the factor at $p=2$+    in CITE{formula-product} is $2$.+  comment-repetitions: By CITE{formula-quadratic}, $C(f)$ depends on $f$ only through+    the character $\chi_\Delta(p)=\left(\frac{\Delta}{p}\right)$. Replacing $\Delta$+    by $\Delta m^2$, where every prime divisor of $m$ already divides $\Delta$, leaves+    this character unchanged; $x^2+1$, $x^2+4$ and $x^2+16$ are one such group.+  comment-conjectural: The Euler product defining $C(f)$ converges conditionally for+    these quadratic polynomials. The asymptotic interpretation as a density of prime+    values remains conjectural.   comment-examples: For $f=x^2+1$, OEIS A199401 gives $C(f)$ CITE{OEISx2plus1}, while     OEIS A331941 gives $C(f)/2$ CITE{OEISx2plus1Half}. For $f=x^2+x+41$, OEIS A221712-    gives $C(f)/2$ CITE{OEISEuler41}.+    gives $C(f)/2$ CITE{OEISEuler41}, and OEIS A331940 lists $41$ as a record addend+    CITE{OEISRecordQuadratics}. Formulas:   formula-product: $C(f)=\prod_p\dfrac{1-N_f(p)/p}{1-1/p}$, where $N_f(p)$ is the     number of roots of $f$ modulo $p$.-  formula-quadratic: If $\Delta=b^2-4c$ for $f=x^2+bx+c$, then $N_f(p)=1+\left(\frac{\Delta}{p}\right)$-    and therefore $C(f)=\prod_p\left(1-\dfrac{1}{p-1}\left(\frac{\Delta}{p}\right)\right)$,-    with the Kronecker symbol $\left(\frac{\Delta}{p}\right)$.+  formula-quadratic: If $\Delta=b^2-4c$ for $f=x^2+bx+c$, then $N_f(p)=1+\left(\frac{\Delta}{p}\right)$,+    so $C(f)=\prod_p\left(1-\dfrac{1}{p-1}\left(\frac{\Delta}{p}\right)\right)$, with+    the Kronecker symbol $\left(\frac{\Delta}{p}\right)$.+  formula-lfunction: With $\chi_\Delta(p)=\left(\frac{\Delta}{p}\right)$ and $L(1,\chi_\Delta)=\prod_p\left(1-\dfrac{\chi_\Delta(p)}{p}\right)^{-1}$,+    $C(f)=\dfrac{1}{L(1,\chi_\Delta)}\prod_p\left(1-\dfrac{\chi_\Delta(p)}{(p-1)(p-\chi_\Delta(p))}\right)$;+    the product in this formula converges absolutely.   formula-asymptotic: The Bateman-Horn conjecture predicts $\#\{n\leq x:f(n)\text{     is prime}\}\sim \dfrac{C(f)}{2}\int_2^x\dfrac{dt}{\log t}$ for the quadratic polynomials-    in this table CITE{WikiBH} CITE{BH1962}.+    in this table CITE{BH1962} CITE{WikiBH}. Programs:-  program-sage:-    language: Sage-    code: 'from generate import BatemanHornQuadraticConstants--      g = BatemanHornQuadraticConstants()--      g.value({''f'': ''x^2+1''}, 100)         # 1.3728134628182460091...--      g.value({''f'': ''x^2+x+41''}, 100)      # 6.6395463549428433306...'   program-pari:     language: PARI/GP-    code: '\\ Load HardyLittlewood2 from the Belabas-Cohen script CITE{BelabasCohenGP}.+    code: '\\ Load https://oeis.org/A221712/a221712.gp.txt, which defines HardyLittlewood2.        default(realprecision, 140)

 - table: HREF{Values_of_Dirichlet_L-functions_at_positive_integers}[Values of Dirichlet     L-functions at positive integers]-  relation: CITE{formula-quadratic} is evaluated through Dirichlet $L$-values of quadratic-    characters+  relation: CITE{formula-lfunction} expresses these constants through Dirichlet $L$-values+    of quadratic characters - table: HREF{Residues_of_Dedekind_zeta_functions_of_quadratic_fields}[Residues of     Dedekind zeta functions of quadratic fields]

       for quadratic polynomials'     url: https://oeis.org/A221712/a221712.gp.txt-  CohenPDF:-    title: 'Henri Cohen: High-precision computation of Hardy-Littlewood constants'-    url: https://oeis.org/A221712/a221712.pdf References:   BH1962:

     the parameter constraint, and every polynomial $x^2+x+a$ with $1\leq a\leq41$     satisfying the parameter constraint-  rigour details: Values were computed with the Belabas-Cohen PARI/GP algorithm CITE{BelabasCohenGP},-    which implements Cohen's accelerated computation of Hardy-Littlewood constants+  rigour details: 'Values were computed with the Belabas-Cohen PARI/GP algorithm CITE{BelabasCohenGP},+    which implements Cohen''s accelerated computation of Hardy-Littlewood constants     CITE{Cohen1998}. PARI was run at $140$ decimal digits, and $100$ digits are written.-    Before the draft was filled, the root-count identity in CITE{formula-quadratic}-    was checked directly modulo every prime $p\leq97$ for every row; $C(x^2+1)$ and-    $C(x^2+x+41)$ were compared with OEIS A199401 CITE{OEISx2plus1} and twice OEIS-    A221712 CITE{OEISEuler41}; the same two values were recomputed with an independent-    Sage character-split zeta expansion; and the generator was rerun at $160$ decimal-    digits on the displayed examples.+++    The root-count identity $N_f(p)=1+\left(\frac{\Delta}{p}\right)$ was checked by+    enumeration for every row and every prime $p\leq97$. The stored value of $C(x^2+1)$+    agrees with OEIS A199401 CITE{OEISx2plus1} and with $2\times$ OEIS A331941 CITE{OEISx2plus1Half};+    the stored value of $C(x^2+x+41)$ agrees with $2\times$ OEIS A221712 CITE{OEISEuler41}.+    Recomputing $C(x^2+1)$ and $C(x^2+x+41)$ at $160$ decimal digits rounds to the+    same $100$ digits.' Display properties:   number-header: $C(f)$ Numbers:-- params:-    f: x^2-30-  number: '0.8601147104299838789007494560328252807806904261844759019415598420694685969143298397526824924379622660'-  comment: Here the polynomial discriminant is $\Delta=120$.-- params:-    f: x^2-29-  number: '1.126772839598820996273857212748371200099365764891258533100946802507352106104383260933955097581403555'-  comment: Here the polynomial discriminant is $\Delta=116$.-- params:-    f: x^2-28-  number: '0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422'-  comment: Here the polynomial discriminant is $\Delta=112$.-- params:-    f: x^2-27-  number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'-  comment: Here the polynomial discriminant is $\Delta=108$.-- params:-    f: x^2-26-  number: '1.168725054064629519927263619162420026682081108668887085636491388421635766085867648737434684009878222'-  comment: Here the polynomial discriminant is $\Delta=104$.-- params:-    f: x^2-24-  number: '1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249'-  comment: Here the polynomial discriminant is $\Delta=96$.-- params:-    f: x^2-23-  number: '1.389165729225091021008641394438937900491782100374239392159940634714728765171873881196703677828836074'-  comment: Here the polynomial discriminant is $\Delta=92$.-- params:-    f: x^2-22-  number: '0.5977866238870616809009077305745767918130060035608807497940575013180175697794438719845945910594857504'-  comment: Here the polynomial discriminant is $\Delta=88$.-- params:-    f: x^2-21-  number: '0.9278845824259630964861938618708887690023142404813744256456850301459903522338283872702898889257291492'-  comment: Here the polynomial discriminant is $\Delta=84$.-- params:-    f: x^2-20-  number: '1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539'-  comment: Here the polynomial discriminant is $\Delta=80$.-- params:-    f: x^2-19-  number: '0.5442381158545459834694417829112003690997867548503895850374429765275947307063046967002063104561103165'-  comment: Here the polynomial discriminant is $\Delta=76$.-- params:-    f: x^2-18-  number: '1.233369606273401136389080676108783664530189797828033202904168142133405692834916133351665887818675312'-  comment: Here the polynomial discriminant is $\Delta=72$.-- params:-    f: x^2-17-  number: '2.360474794680180004013217069970078663275795672137371417024644858094859086906230212900083312385861010'-  comment: Here the polynomial discriminant is $\Delta=68$.-- params:-    f: x^2-15-  number: '0.9117194099555348423618460572171063721199634074407231727580630198006585349049350488326141429725867783'-  comment: Here the polynomial discriminant is $\Delta=60$.-- params:-    f: x^2-14-  number: '1.151675729617594721217200488222297587405377081685228924691542287580788442705518760485501579275648485'-  comment: Here the polynomial discriminant is $\Delta=56$.-- params:-    f: x^2-13-  number: '0.8072362275555245670688831014666742044407393694788056234907708193950660811802836289826826609765204381'-  comment: Here the polynomial discriminant is $\Delta=52$.-- params:-    f: x^2-12-  number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'-  comment: Here the polynomial discriminant is $\Delta=48$.-- params:-    f: x^2-11-  number: '1.147979957061047562456689725537627701501502268797270468461706803083155724182282459186860059680031253'-  comment: Here the polynomial discriminant is $\Delta=44$.-- params:-    f: x^2-10-  number: '0.6711139154474606895516481167249919229589063161426530994908947791974688362715536026551029106420576270'-  comment: Here the polynomial discriminant is $\Delta=40$.-- params:-    f: x^2-8-  number: '1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968'-  comment: Here the polynomial discriminant is $\Delta=32$.-- params:-    f: x^2-7-  number: '0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422'-  comment: Here the polynomial discriminant is $\Delta=28$.-- params:-    f: x^2-6-  number: '1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249'-  comment: Here the polynomial discriminant is $\Delta=24$.-- params:-    f: x^2-5-  number: '1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539'-  comment: Here the polynomial discriminant is $\Delta=20$.-- params:-    f: x^2-3-  number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'-  comment: Here the polynomial discriminant is $\Delta=12$.-- params:-    f: x^2-2-  number: '1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968'-  comment: Here the polynomial discriminant is $\Delta=8$.-- params:-    f: x^2+1-  number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'-  comment: Here the polynomial discriminant is $\Delta=-4$; OEIS A199401 gives this-    normalization and A331941 gives half of it.-- params:-    f: x^2+2-  number: '0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792'-  comment: Here the polynomial discriminant is $\Delta=-8$.-- params:-    f: x^2+3-  number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'-  comment: Here the polynomial discriminant is $\Delta=-12$.-- params:-    f: x^2+4-  number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'-  comment: Here the polynomial discriminant is $\Delta=-16$.-- params:-    f: x^2+5-  number: '0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139'-  comment: Here the polynomial discriminant is $\Delta=-20$.-- params:-    f: x^2+6-  number: '0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893'-  comment: Here the polynomial discriminant is $\Delta=-24$.-- params:-    f: x^2+7-  number: '1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028'-  comment: Here the polynomial discriminant is $\Delta=-28$.-- params:-    f: x^2+8-  number: '0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792'-  comment: Here the polynomial discriminant is $\Delta=-32$.-- params:-    f: x^2+9-  number: '0.9152089752121640060747951311513459121188887341705063719574894315518948178115801544851465379593919892'-  comment: Here the polynomial discriminant is $\Delta=-36$.-- params:-    f: x^2+10-  number: '1.082290322622870934008726464648342274750746063264927783711232919691927699363829647117420572025543324'-  comment: Here the polynomial discriminant is $\Delta=-40$.-- params:-    f: x^2+11-  number: '0.5101385751188618061860056763665242252404661892233736103082430377948040609666054252689766154191079408'-  comment: Here the polynomial discriminant is $\Delta=-44$.-- params:-    f: x^2+12-  number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'-  comment: Here the polynomial discriminant is $\Delta=-48$.-- params:-    f: x^2+13-  number: '1.285787607594973099821388187755306836651177012482711025958632819412813134808814396163949191153115206'-  comment: Here the polynomial discriminant is $\Delta=-52$.-- params:-    f: x^2+14-  number: '0.4203685120242415850612291867691435857669957869166042938990290458429773598292241811053479296717706134'-  comment: Here the polynomial discriminant is $\Delta=-56$.-- params:-    f: x^2+15-  number: '1.267019954790685681315350247382006105145768371868654178910125512950871437236030883609580774364585537'-  comment: Here the polynomial discriminant is $\Delta=-60$.-- params:-    f: x^2+16-  number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'-  comment: Here the polynomial discriminant is $\Delta=-64$.-- params:-    f: x^2+17-  number: '0.4917097430948072341233640382833226389861633259260834511662290709132051750511550610602398378623044735'-  comment: Here the polynomial discriminant is $\Delta=-68$.-- params:-    f: x^2+18-  number: '1.426126208480385980446525214799653675136896481678451332214802405299149841356752800235828337740280558'-  comment: Here the polynomial discriminant is $\Delta=-72$.-- params:-    f: x^2+19-  number: '0.9422204577474599205433576755296373799993034647265656299752433984881648313597146577902812948518917527'-  comment: Here the polynomial discriminant is $\Delta=-76$.-- params:-    f: x^2+20-  number: '0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139'-  comment: Here the polynomial discriminant is $\Delta=-80$.-- params:-    f: x^2+21-  number: '0.6754967305699162440248975466908905318610594026999808150795609624059682995907108895254707935118439946'-  comment: Here the polynomial discriminant is $\Delta=-84$.-- params:-    f: x^2+22-  number: '1.766762995152103665725714568256978814731409229751309691306775260082484003131677025944646277449270985'-  comment: Here the polynomial discriminant is $\Delta=-88$.-- params:-    f: x^2+23-  number: '0.8166075041323586681283113818893528970340129178951570930204595471408835704259581029553830684300264873'-  comment: Here the polynomial discriminant is $\Delta=-92$.-- params:-    f: x^2+24-  number: '0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893'-  comment: Here the polynomial discriminant is $\Delta=-96$.-- params:-    f: x^2+25-  number: '1.830417950424328012149590262302691824237777468341012743914978863103789635623160308970293075918783978'-  comment: Here the polynomial discriminant is $\Delta=-100$.-- params:-    f: x^2+26-  number: '0.3733514180011663008517026122804559993013766152391805932889147796919352213980823203194354170039387262'-  comment: Here the polynomial discriminant is $\Delta=-104$.-- params:-    f: x^2+27-  number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'-  comment: Here the polynomial discriminant is $\Delta=-108$.-- params:-    f: x^2+28-  number: '1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028'-  comment: Here the polynomial discriminant is $\Delta=-112$.-- params:-    f: x^2+29-  number: '0.4042724938761413201705112848081949197798504701269073648328608973002068055601973728845968281095300364'-  comment: Here the polynomial discriminant is $\Delta=-116$.-- params:-    f: x^2+30-  number: '0.8695751905453238912022643067607602377970713513389461248511546547987457009790152536448111667612905273'-  comment: Here the polynomial discriminant is $\Delta=-120$.-- params:-    f: x^2+x+1-  number: '2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736'-  comment: Here the polynomial discriminant is $\Delta=-3$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+3-  number: '1.020277150237723612372011352733048450480932378446747220616486075589608121933210850537953230838215882'-  comment: Here the polynomial discriminant is $\Delta=-11$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+5-  number: '1.884440915494919841086715351059274759998606929453131259950486796976329662719429315580562589703783505'-  comment: Here the polynomial discriminant is $\Delta=-19$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+7-  number: '2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736'-  comment: Here the polynomial discriminant is $\Delta=-27$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+9-  number: '0.9280103731273487855873097252803947565787508509905504594844853433869593978188827344697684992905128316'-  comment: Here the polynomial discriminant is $\Delta=-35$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+11-  number: '3.259441847518388145093386059923914196725620715892583498750020991085655587697066611871551227021417497'-  comment: Here the polynomial discriminant is $\Delta=-43$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+13-  number: '1.419857358966315738255882478053358602981063388340692803832065566567952327363225826438664468845011582'-  comment: Here the polynomial discriminant is $\Delta=-51$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+15-  number: '0.7506330693483119384816695337893102809050716260116440015777356381958311886817350455720476745762888187'-  comment: Here the polynomial discriminant is $\Delta=-59$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+17-  number: '4.174661613064506156384616027265554205258359532355780762832290173805074233803615278659632440732471028'-  comment: Here the polynomial discriminant is $\Delta=-67$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+19-  number: '1.793172405678866224989334697976699546821688129793060290843064738170249083475277042217185132184009389'-  comment: Here the polynomial discriminant is $\Delta=-75$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+21-  number: '0.9700198069727936513487153646506794009553303496568042714539174121546204557329979833218089012583478501'-  comment: Here the polynomial discriminant is $\Delta=-83$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+23-  number: '2.141301828834744177405561293675416841044042531319356004932445116309036103158136828498402877268236243'-  comment: Here the polynomial discriminant is $\Delta=-91$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+25-  number: '2.040554300475447224744022705466096900961864756893494441232972151179216243866421701075906461676431763'-  comment: Here the polynomial discriminant is $\Delta=-99$; the local factor at $p=2$-    is $2$.-- params:-    f: x^2+x+27-  number: '1.143878070461191914460965968416389087615849533015381762745042519987791242580107196598246225550931869'-  comment: Here the polynomial discriminant is $\Delta=-107$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+29-  number: '2.509924528638309698164020770571015618012214936531440762279479806568444102037061777052853261363088714'-  comment: Here the polynomial discriminant is $\Delta=-115$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+31-  number: '2.482812754958659435877802042265728699767214193716973768198429923255571125959857518638268319299046271'-  comment: Here the polynomial discriminant is $\Delta=-123$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+33-  number: '0.6596657567089936838732409315054822272005742886441638363420171151419848490461618510829897723956105335'-  comment: Here the polynomial discriminant is $\Delta=-131$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+35-  number: '1.689602419947677057250855974880387107256813527569778238330400896888306148814092238563942505797773478'-  comment: Here the polynomial discriminant is $\Delta=-139$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+37-  number: '2.689758608518299337484002046965049320232532194689590436264597107255373625212915563325777698276014083'-  comment: Here the polynomial discriminant is $\Delta=-147$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+39-  number: '1.003074745695317297913893711663044683123966880363528175911436776511523502224541451536167360622096167'-  comment: Here the polynomial discriminant is $\Delta=-155$; the local factor at-    $p=2$ is $2$.-- params:-    f: x^2+x+41-  number: '6.639546354942843330647113715299775932937109171305971698307881455900526620852236299475102289608912434'-  comment: Here the polynomial discriminant is $\Delta=-163$; the local factor at-    $p=2$ is $2$; OEIS A221712 gives half of this normalization.+  x^2-30:+    number: '0.8601147104299838789007494560328252807806904261844759019415598420694685969143298397526824924379622660'+    comment: The discriminant is $\Delta=120$.+  x^2-29:+    number: '1.126772839598820996273857212748371200099365764891258533100946802507352106104383260933955097581403555'+    comment: The discriminant is $\Delta=116$.+  x^2-28:+    number: '0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422'+    comment: The discriminant is $\Delta=112$.+  x^2-27:+    number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'+    comment: The discriminant is $\Delta=108$.+  x^2-26:+    number: '1.168725054064629519927263619162420026682081108668887085636491388421635766085867648737434684009878222'+    comment: The discriminant is $\Delta=104$.+  x^2-24:+    number: '1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249'+    comment: The discriminant is $\Delta=96$.+  x^2-23:+    number: '1.389165729225091021008641394438937900491782100374239392159940634714728765171873881196703677828836074'+    comment: The discriminant is $\Delta=92$.+  x^2-22:+    number: '0.5977866238870616809009077305745767918130060035608807497940575013180175697794438719845945910594857504'+    comment: The discriminant is $\Delta=88$.+  x^2-21:+    number: '0.9278845824259630964861938618708887690023142404813744256456850301459903522338283872702898889257291492'+    comment: The discriminant is $\Delta=84$.+  x^2-20:+    number: '1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539'+    comment: The discriminant is $\Delta=80$.+  x^2-19:+    number: '0.5442381158545459834694417829112003690997867548503895850374429765275947307063046967002063104561103165'+    comment: The discriminant is $\Delta=76$.+  x^2-18:+    number: '1.233369606273401136389080676108783664530189797828033202904168142133405692834916133351665887818675312'+    comment: The discriminant is $\Delta=72$.+  x^2-17:+    number: '2.360474794680180004013217069970078663275795672137371417024644858094859086906230212900083312385861010'+    comment: The discriminant is $\Delta=68$.+  x^2-15:+    number: '0.9117194099555348423618460572171063721199634074407231727580630198006585349049350488326141429725867783'+    comment: The discriminant is $\Delta=60$.+  x^2-14:+    number: '1.151675729617594721217200488222297587405377081685228924691542287580788442705518760485501579275648485'+    comment: The discriminant is $\Delta=56$.+  x^2-13:+    number: '0.8072362275555245670688831014666742044407393694788056234907708193950660811802836289826826609765204381'+    comment: The discriminant is $\Delta=52$.+  x^2-12:+    number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'+    comment: The discriminant is $\Delta=48$.+  x^2-11:+    number: '1.147979957061047562456689725537627701501502268797270468461706803083155724182282459186860059680031253'+    comment: The discriminant is $\Delta=44$.+  x^2-10:+    number: '0.6711139154474606895516481167249919229589063161426530994908947791974688362715536026551029106420576270'+    comment: The discriminant is $\Delta=40$.+  x^2-8:+    number: '1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968'+    comment: The discriminant is $\Delta=32$.+  x^2-7:+    number: '0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422'+    comment: The discriminant is $\Delta=28$.+  x^2-6:+    number: '1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249'+    comment: The discriminant is $\Delta=24$.+  x^2-5:+    number: '1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539'+    comment: The discriminant is $\Delta=20$.+  x^2-3:+    number: '1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473'+    comment: The discriminant is $\Delta=12$.+  x^2-2:+    number: '1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968'+    comment: The discriminant is $\Delta=8$.+  x^2+1:+    number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'+    comment: The discriminant is $\Delta=-4$; primes of the form $n^2+1$ are Landau's+      fourth problem CITE{WikiLandau}; OEIS A199401 gives this normalization, A331941+      gives half of it, and A206709 counts such primes CITE{OEISCountsX2Plus1}.+  x^2+2:+    number: '0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792'+    comment: The discriminant is $\Delta=-8$.+  x^2+3:+    number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'+    comment: The discriminant is $\Delta=-12$.+  x^2+4:+    number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'+    comment: The discriminant is $\Delta=-16$.+  x^2+5:+    number: '0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139'+    comment: The discriminant is $\Delta=-20$.+  x^2+6:+    number: '0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893'+    comment: The discriminant is $\Delta=-24$.+  x^2+7:+    number: '1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028'+    comment: The discriminant is $\Delta=-28$.+  x^2+8:+    number: '0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792'+    comment: The discriminant is $\Delta=-32$.+  x^2+9:+    number: '0.9152089752121640060747951311513459121188887341705063719574894315518948178115801544851465379593919892'+    comment: The discriminant is $\Delta=-36$.+  x^2+10:+    number: '1.082290322622870934008726464648342274750746063264927783711232919691927699363829647117420572025543324'+    comment: The discriminant is $\Delta=-40$.+  x^2+11:+    number: '0.5101385751188618061860056763665242252404661892233736103082430377948040609666054252689766154191079408'+    comment: The discriminant is $\Delta=-44$.+  x^2+12:+    number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'+    comment: The discriminant is $\Delta=-48$.+  x^2+13:+    number: '1.285787607594973099821388187755306836651177012482711025958632819412813134808814396163949191153115206'+    comment: The discriminant is $\Delta=-52$.+  x^2+14:+    number: '0.4203685120242415850612291867691435857669957869166042938990290458429773598292241811053479296717706134'+    comment: The discriminant is $\Delta=-56$.+  x^2+15:+    number: '1.267019954790685681315350247382006105145768371868654178910125512950871437236030883609580774364585537'+    comment: The discriminant is $\Delta=-60$.+  x^2+16:+    number: '1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984'+    comment: The discriminant is $\Delta=-64$.+  x^2+17:+    number: '0.4917097430948072341233640382833226389861633259260834511662290709132051750511550610602398378623044735'+    comment: The discriminant is $\Delta=-68$.+  x^2+18:+    number: '1.426126208480385980446525214799653675136896481678451332214802405299149841356752800235828337740280558'+    comment: The discriminant is $\Delta=-72$.+  x^2+19:+    number: '0.9422204577474599205433576755296373799993034647265656299752433984881648313597146577902812948518917527'+    comment: The discriminant is $\Delta=-76$.+  x^2+20:+    number: '0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139'+    comment: The discriminant is $\Delta=-80$.+  x^2+21:+    number: '0.6754967305699162440248975466908905318610594026999808150795609624059682995907108895254707935118439946'+    comment: The discriminant is $\Delta=-84$.+  x^2+22:+    number: '1.766762995152103665725714568256978814731409229751309691306775260082484003131677025944646277449270985'+    comment: The discriminant is $\Delta=-88$.+  x^2+23:+    number: '0.8166075041323586681283113818893528970340129178951570930204595471408835704259581029553830684300264873'+    comment: The discriminant is $\Delta=-92$.+  x^2+24:+    number: '0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893'+    comment: The discriminant is $\Delta=-96$.+  x^2+25:+    number: '1.830417950424328012149590262302691824237777468341012743914978863103789635623160308970293075918783978'+    comment: The discriminant is $\Delta=-100$.+  x^2+26:+    number: '0.3733514180011663008517026122804559993013766152391805932889147796919352213980823203194354170039387262'+    comment: The discriminant is $\Delta=-104$.+  x^2+27:+    number: '1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868'+    comment: The discriminant is $\Delta=-108$.+  x^2+28:+    number: '1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028'+    comment: The discriminant is $\Delta=-112$.+  x^2+29:+    number: '0.4042724938761413201705112848081949197798504701269073648328608973002068055601973728845968281095300364'+    comment: The discriminant is $\Delta=-116$.+  x^2+30:+    number: '0.8695751905453238912022643067607602377970713513389461248511546547987457009790152536448111667612905273'+    comment: The discriminant is $\Delta=-120$.+  x^2+x+1:+    number: '2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736'+    comment: The discriminant is $\Delta=-3$.+  x^2+x+3:+    number: '1.020277150237723612372011352733048450480932378446747220616486075589608121933210850537953230838215882'+    comment: The discriminant is $\Delta=-11$.+  x^2+x+5:+    number: '1.884440915494919841086715351059274759998606929453131259950486796976329662719429315580562589703783505'+    comment: The discriminant is $\Delta=-19$.+  x^2+x+7:+    number: '2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736'+    comment: The discriminant is $\Delta=-27$.+  x^2+x+9:+    number: '0.9280103731273487855873097252803947565787508509905504594844853433869593978188827344697684992905128316'+    comment: The discriminant is $\Delta=-35$.+  x^2+x+11:+    number: '3.259441847518388145093386059923914196725620715892583498750020991085655587697066611871551227021417497'+    comment: The discriminant is $\Delta=-43$.+  x^2+x+13:+    number: '1.419857358966315738255882478053358602981063388340692803832065566567952327363225826438664468845011582'+    comment: The discriminant is $\Delta=-51$.+  x^2+x+15:+    number: '0.7506330693483119384816695337893102809050716260116440015777356381958311886817350455720476745762888187'+    comment: The discriminant is $\Delta=-59$.+  x^2+x+17:+    number: '4.174661613064506156384616027265554205258359532355780762832290173805074233803615278659632440732471028'+    comment: The discriminant is $\Delta=-67$.+  x^2+x+19:+    number: '1.793172405678866224989334697976699546821688129793060290843064738170249083475277042217185132184009389'+    comment: The discriminant is $\Delta=-75$.+  x^2+x+21:+    number: '0.9700198069727936513487153646506794009553303496568042714539174121546204557329979833218089012583478501'+    comment: The discriminant is $\Delta=-83$.+  x^2+x+23:+    number: '2.141301828834744177405561293675416841044042531319356004932445116309036103158136828498402877268236243'+    comment: The discriminant is $\Delta=-91$.+  x^2+x+25:+    number: '2.040554300475447224744022705466096900961864756893494441232972151179216243866421701075906461676431763'+    comment: The discriminant is $\Delta=-99$.+  x^2+x+27:+    number: '1.143878070461191914460965968416389087615849533015381762745042519987791242580107196598246225550931869'+    comment: The discriminant is $\Delta=-107$.+  x^2+x+29:+    number: '2.509924528638309698164020770571015618012214936531440762279479806568444102037061777052853261363088714'+    comment: The discriminant is $\Delta=-115$.+  x^2+x+31:+    number: '2.482812754958659435877802042265728699767214193716973768198429923255571125959857518638268319299046271'+    comment: The discriminant is $\Delta=-123$.+  x^2+x+33:+    number: '0.6596657567089936838732409315054822272005742886441638363420171151419848490461618510829897723956105335'+    comment: The discriminant is $\Delta=-131$.+  x^2+x+35:+    number: '1.689602419947677057250855974880387107256813527569778238330400896888306148814092238563942505797773478'+    comment: The discriminant is $\Delta=-139$.+  x^2+x+37:+    number: '2.689758608518299337484002046965049320232532194689590436264597107255373625212915563325777698276014083'+    comment: The discriminant is $\Delta=-147$.+  x^2+x+39:+    number: '1.003074745695317297913893711663044683123966880363528175911436776511523502224541451536167360622096167'+    comment: The discriminant is $\Delta=-155$.+  x^2+x+41:+    number: '6.639546354942843330647113715299775932937109171305971698307881455900526620852236299475102289608912434'+    comment: The discriminant is $\Delta=-163$; OEIS A221712 gives half of this normalization,+      and OEIS A331940 lists $41$ as a record addend CITE{OEISRecordQuadratics}.