Salem numbers less than 1.3

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Numbers

coefficients

$\tau$

1, 1, 0, -1, -1, -1:

1.176280818259917506544070338474035050693415806564695259830106347029688376548549962096830115581815395

comment: This Salem number has degree $10$, with minimal polynomial $x^{10} + x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + x + 1$. It is Lehmer's number and the growth rate of the Coxeter triangle group $\Delta(2,3,7)$.

1, -1, 1, -1, 0, 0, -1, 1, -1, 1:

1.188368147508223588142960958629593594704704560062905688741453371291960641402174589475835098806796775

comment: degree $18$; minimal polynomial $x^{18} - x^{17} + x^{16} - x^{15} - x^{12} + x^{11} - x^{10} + x^{9} - x^{8} + x^{7} - x^{6} - x^{3} + x^{2} - x + 1$.

1, 0, 0, -1, -1, 0, 0, 1:

1.200026523987391518902962100414601567240618151999851067924399839886073113442524096442461727884969910

comment: degree $14$; minimal polynomial $x^{14} - x^{11} - x^{10} + x^{7} - x^{4} - x^{3} + 1$.

1, 0, -1, 0, 0, 0, 0, -1:

1.202616743688604261118295415948619045343949834969523043685309576726454065876365553772671080055182658

comment: degree $14$; minimal polynomial $x^{14} - x^{12} - x^{7} - x^{2} + 1$.

1, 0, 0, 0, -1, -1:

1.216391661138265091626806311199463327722253606570570757560427065838312129461849594426379666709543954

comment: degree $10$; minimal polynomial $x^{10} - x^{6} - x^{5} - x^{4} + 1$.

1, -1, 0, 0, 0, 0, 0, 0, -1, 1:

1.219720859040311844169606760414677944390415505541569678287974417873384645990839065835539320785162596

comment: degree $18$; minimal polynomial $x^{18} - x^{17} - x^{10} + x^{9} - x^{8} - x + 1$.

1, 0, 0, -1, 0, -1:

1.230391434407224702790177938975279017566574489661756241401914236172813447853545416735984651662408529

comment: This Salem number has degree $10$, with minimal polynomial $x^{10} - x^{7} - x^{5} - x^{3} + 1$. It is the growth rate of the Coxeter triangle group $\Delta(2,3,8)$.

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1:

1.232613548593121003962731694807909791411577371209831046729916582053839351373895703134978488530324922

comment: degree $20$; minimal polynomial $x^{20} - x^{19} - x^{15} + x^{14} - x^{11} + x^{10} - x^{9} + x^{6} - x^{5} - x + 1$.

1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1, -1:

1.235664580389747308105169351531263479723510042746239077650438077206311924940278149889166074042638593

comment: degree $22$; minimal polynomial $x^{22} - x^{20} - x^{19} + x^{15} + x^{14} - x^{12} - x^{11} - x^{10} + x^{8} + x^{7} - x^{3} - x^{2} + 1$.

1, -1, 0, 0, 0, 0, 0, 0, -1:

1.236317931803230489899094869802054553944819208367869563794753784111836999567141563427243854365405562

comment: degree $16$; minimal polynomial $x^{16} - x^{15} - x^{8} - x + 1$.

1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1:

1.237504821217490608171021829989092378856326086862049849963623164314795149153713239330116879323681527

comment: degree $26$; minimal polynomial $x^{26} - x^{24} - x^{21} - x^{18} + x^{16} + x^{13} + x^{10} - x^{8} - x^{5} - x^{2} + 1$.

1, -1, 1, -1, 0, 0, -1:

1.240726423652541392056148161575665656805788554669818945154505874492081494833373347611149564568350540

comment: degree $12$; minimal polynomial $x^{12} - x^{11} + x^{10} - x^{9} - x^{6} - x^{3} + x^{2} - x + 1$.

1, 0, 0, 0, 0, 0, -1, -1, -1, -1:

1.252775937410113900864582824053204309304769854195780259121070767220447361924730151351933663878789295

comment: degree $18$; minimal polynomial $x^{18} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} + 1$.

1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0:

1.253330650201489757028162788986138235892342419486482436157483570794348084059380473979534453277009941

comment: degree $20$; minimal polynomial $x^{20} - x^{18} - x^{15} - x^{5} - x^{2} + 1$.

1, 0, -1, -1, 0, 1, 0, -1:

1.255093516763722879173091003232857070130650899913269851336588823459118544428340951581671250107775401

comment: degree $14$; minimal polynomial $x^{14} - x^{12} - x^{11} + x^{9} - x^{7} + x^{5} - x^{3} - x^{2} + 1$.

1, -1, 0, 0, -1, 1, 0, 0, 0, -1:

1.256221154391670233067434043309669163943842183894362968698551997499450681251721148671121281918681872

comment: degree $18$; minimal polynomial $x^{18} - x^{17} - x^{14} + x^{13} - x^{9} + x^{5} - x^{4} - x + 1$.

1, -1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 1, -1:

1.260103540354990920321649852331678168961999664904333174814990287796601346085108215017984119208171229

comment: degree $24$; minimal polynomial $x^{24} - x^{23} - x^{20} + x^{19} - x^{17} + x^{16} - x^{15} + x^{13} - x^{12} + x^{11} - x^{9} + x^{8} - x^{7} + x^{5} - x^{4} - x + 1$.

1, -1, 0, -1, 1, 0, 0, 0, -1, 1, -1, 1:

1.260284236896492963739228435283866365638551287842634180725403343395901937301970465990247417973123276

comment: degree $22$; minimal polynomial $x^{22} - x^{21} - x^{19} + x^{18} - x^{14} + x^{13} - x^{12} + x^{11} - x^{10} + x^{9} - x^{8} + x^{4} - x^{3} - x + 1$.

1, 0, -1, 0, 0, -1:

1.261230961137138851946671503074483335906561512823932899174657356830904239469104488591116935467487797

comment: This Salem number has degree $10$, with minimal polynomial $x^{10} - x^{8} - x^{5} - x^{2} + 1$. It is the growth rate of the Coxeter triangle group $\Delta(2,3,9)$.

1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1:

1.263038139930169261222022798085163082403989294603617622448883468909444891211816578542249677559000514

comment: degree $26$; minimal polynomial $x^{26} - x^{25} - x^{20} + x^{13} - x^{6} - x + 1$.

1, -1, 0, 0, 0, 0, -1, 1:

1.267296442523068692734077407604822492629074176701971734433405054449451752853913983609519547785241871

comment: degree $14$; minimal polynomial $x^{14} - x^{13} - x^{8} + x^{7} - x^{6} - x + 1$.

1, -1, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1:

1.276779674019016861136497157605067169395425247666261436185614595924285532748953111077682351101737238

comment: degree $22$; minimal polynomial $x^{22} - x^{21} - x^{20} + x^{19} - x^{13} + x^{11} - x^{9} + x^{3} - x^{2} - x + 1$.

1, 0, 0, -1, -1:

1.280638156267757596701902532710676301595313317819353584076277261716335555633386491878399947424007022

comment: This Salem number has degree $8$, with minimal polynomial $x^{8} - x^{5} - x^{4} - x^{3} + 1$. It is the growth rate of the Coxeter triangle groups $\Delta(2,3,10)$ and $\Delta(2,4,5)$.

1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1:

1.281691371528106310055107748672771558415642455515779917413082415055377327487741807397084621153512449

comment: degree $26$; minimal polynomial $x^{26} - x^{20} - x^{19} - x^{18} - x^{17} - x^{16} - x^{15} - x^{14} - x^{13} - x^{12} - x^{11} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} + 1$.

1, -2, 2, -2, 2, -2, 1, 0, -1, 1, -1:

1.282495560639960169561207128806572472485581139082169212292401024502310661104427644540700765924178039

comment: degree $20$; minimal polynomial $x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 2 x^{16} - 2 x^{15} + x^{14} - x^{12} + x^{11} - x^{10} + x^{9} - x^{8} + x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 2 x^{2} - 2 x + 1$.

1, 0, 0, 0, -1, 0, -1, -1, 0, -1:

1.284616550925536736743131441485789494939191059613661928484866385712738555102877745989023559495027088

comment: degree $18$; minimal polynomial $x^{18} - x^{14} - x^{12} - x^{11} - x^{9} - x^{7} - x^{6} - x^{4} + 1$.

1, -2, 1, 1, -2, 1, 0, 0, -1, 1, 0, -1, 1, -1:

1.284746821544843035729838140650008200718705398881565768980866336566720406099103932587188672386882989

comment: degree $26$; minimal polynomial $x^{26} - 2 x^{25} + x^{24} + x^{23} - 2 x^{22} + x^{21} - x^{18} + x^{17} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{9} - x^{8} + x^{5} - 2 x^{4} + x^{3} + x^{2} - 2 x + 1$.

1, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1:

1.285099363651876557117420034512953144874203997952486708227935530217761432981905301308232456428021224

comment: degree $30$; minimal polynomial $x^{30} - x^{25} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} + x^{15} - x^{10} - x^{9} - x^{8} - x^{7} - x^{6} - x^{5} + 1$.

1, -2, 2, -2, 1, 0, -1, 2, -2, 1, 0, -1, 1, -1, 1, -1:

1.285121520153207532780681369174516237613271731378863673632885847118882541168037735375242283387422682

comment: degree $30$; minimal polynomial $x^{30} - 2 x^{29} + 2 x^{28} - 2 x^{27} + x^{26} - x^{24} + 2 x^{23} - 2 x^{22} + x^{21} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{9} - 2 x^{8} + 2 x^{7} - x^{6} + x^{4} - 2 x^{3} + 2 x^{2} - 2 x + 1$.

1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1:

1.285185670752909791356387310393076486777398243338615604769858775841299139513109128813811005825606248

comment: degree $30$; minimal polynomial $x^{30} - x^{29} - x^{22} - x^{18} - x^{15} - x^{12} - x^{8} - x + 1$.

1, 0, -1, -1, 0, 0, 0, 1, 0, -1, -1, 0, 1, 1:

1.285196726769853432068127270005288448365700170442190244716562984645683818344535826284176377237841781

comment: degree $26$; minimal polynomial $x^{26} - x^{24} - x^{23} + x^{19} - x^{17} - x^{16} + x^{14} + x^{13} + x^{12} - x^{10} - x^{9} + x^{7} - x^{3} - x^{2} + 1$.

1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1:

1.285199179205612167312192383918896566508760671530769225048432083986362622524512169843198828884513257

comment: degree $44$; minimal polynomial $x^{44} - x^{43} - x^{37} - x^{33} + x^{25} + x^{22} + x^{19} - x^{11} - x^{7} - x + 1$.

1, 0, -1, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 1, 0, -1:

1.285235436228923770828415884707550856833387916488331043924772375325234413835111143450140123932228167

comment: degree $30$; minimal polynomial $x^{30} - x^{28} - x^{25} - x^{24} + x^{20} + x^{17} - x^{15} + x^{13} + x^{10} - x^{6} - x^{5} - x^{2} + 1$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1:

1.285409064765363764030309848277292740660005468089094281561805724703087406277172378851538143095670345

comment: degree $34$; minimal polynomial $x^{34} - x^{33} - x^{30} + x^{29} - x^{28} + x^{26} - x^{25} + x^{24} - x^{22} + x^{21} - x^{20} + x^{18} - x^{17} + x^{16} - x^{14} + x^{13} - x^{12} + x^{10} - x^{9} + x^{8} - x^{6} + x^{5} - x^{4} - x + 1$.

1, -2, 2, -2, 2, -2, 2, -3, 3, -3:

1.286395966836277224044411092745880239115725375942247231746941191653717183126508052297978028380193482

comment: degree $18$; minimal polynomial $x^{18} - 2 x^{17} + 2 x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} + 2 x^{12} - 3 x^{11} + 3 x^{10} - 3 x^{9} + 3 x^{8} - 3 x^{7} + 2 x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 2 x^{2} - 2 x + 1$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1:

1.286730182048201274368747282841750891759038499111820286542038482649973354992548669426849319436738804

comment: degree $26$; minimal polynomial $x^{26} - x^{25} - x^{22} + x^{21} - x^{20} + x^{18} - x^{17} + x^{16} - x^{14} + x^{13} - x^{12} + x^{10} - x^{9} + x^{8} - x^{6} + x^{5} - x^{4} - x + 1$.

1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0:

1.291741425714500483635599066106538616717832154961085041460330006512822156937195506601747818535777278

comment: degree $24$; minimal polynomial $x^{24} - x^{23} - x^{18} - x^{6} - x + 1$.

1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1:

1.292039106017929461943480560567260344484221751942270774576911401536994669762487376427933939047638425

comment: degree $20$; minimal polynomial $x^{20} - x^{18} - x^{15} - x^{12} + x^{10} - x^{8} - x^{5} - x^{2} + 1$.

1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 1:

1.292418657582426546281031229140237872352241309398852816864941224955073998776387882342796034382262005

comment: This Salem number has degree $40$, with minimal polynomial $x^{40} - x^{37} - x^{35} - x^{33} - x^{31} - x^{29} + x^{26} + x^{24} + x^{22} + x^{20} + x^{18} + x^{16} + x^{14} - x^{11} - x^{9} - x^{7} - x^{5} - x^{3} + 1$. It is one of four small Salem numbers discovered by Mossinghoff [4].

1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1:

1.292900721780102794630870596243860725721161208584711383482200434962859185172199055129104927356620452

comment: This Salem number has degree $46$, with minimal polynomial $x^{46} - x^{42} - x^{41} - x^{40} - x^{39} + x^{25} + x^{24} + x^{23} + x^{22} + x^{21} - x^{7} - x^{6} - x^{5} - x^{4} + 1$. It is the only entry in Mossinghoff's list with degree greater than $44$. It is one of four small Salem numbers discovered by Mossinghoff [4].

1, 0, -1, -1, 0, 1:

1.293485953125454106519909883794095300396496621900788800582481240648323409236024590193726706779689885

comment: This Salem number has degree $10$, with minimal polynomial $x^{10} - x^{8} - x^{7} + x^{5} - x^{3} - x^{2} + 1$. It is the growth rate of the Coxeter triangle group $\Delta(2,3,11)$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1:

1.295675371944048235295741653561740294973968668937571897650122796909196596141953597496510031118860419

comment: degree $18$; minimal polynomial $x^{18} - x^{17} - x^{14} + x^{13} - x^{12} + x^{10} - x^{9} + x^{8} - x^{6} + x^{5} - x^{4} - x + 1$.

1, -1, 0, -1, 0, 1, 0, 1, -2, 0, 0, 1, 1, -1, -1, -1, 1, 1:

1.296210659593309216851783179125375404230723736392617683646341971540035750766355537270046081016225984

comment: This Salem number has degree $34$, with minimal polynomial $x^{34} - x^{33} - x^{31} + x^{29} + x^{27} - 2 x^{26} + x^{23} + x^{22} - x^{21} - x^{20} - x^{19} + x^{18} + x^{17} + x^{16} - x^{15} - x^{14} - x^{13} + x^{12} + x^{11} - 2 x^{8} + x^{7} + x^{5} - x^{3} - x + 1$. It is one of four small Salem numbers discovered by Mossinghoff [4].

1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1:

1.296421365194547218873224266498702275589617222774875775383982464169375320434412925688408480615942763

comment: degree $22$; minimal polynomial $x^{22} - x^{21} - x^{17} + x^{11} - x^{5} - x + 1$.

1, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 1, 1, 1:

1.296821373714950077456125855369735994986444530041748134229200029748496227132255961439367373333236486

comment: degree $28$; minimal polynomial $x^{28} - x^{24} - x^{23} - x^{22} - x^{21} - x^{20} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} - x^{8} - x^{7} - x^{6} - x^{5} - x^{4} + 1$.

1, 1, 0, -1, -2, -2, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1:

1.298429835475111538327805425740275196702460146064929341491216341055096941191685431457263595600737868

comment: This Salem number has degree $36$, with minimal polynomial $x^{36} + x^{35} - x^{33} - 2 x^{32} - 2 x^{31} - x^{30} + x^{28} + x^{27} - x^{25} - x^{24} + x^{22} + x^{21} - x^{19} - x^{18} - x^{17} + x^{15} + x^{14} - x^{12} - x^{11} + x^{9} + x^{8} - x^{6} - 2 x^{5} - 2 x^{4} - x^{3} + x + 1$. It is one of four small Salem numbers discovered by Mossinghoff [4].

1, -1, -1, 0, 2, 0, -2, -1, 2, 2, -2, -2, 0, 3:

1.299744869472170731620096386139687307907471991371031488761566141077614801002645724407169725316284758

comment: degree $26$; minimal polynomial $x^{26} - x^{25} - x^{24} + 2 x^{22} - 2 x^{20} - x^{19} + 2 x^{18} + 2 x^{17} - 2 x^{16} - 2 x^{15} + 3 x^{13} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 2 x^{8} - x^{7} - 2 x^{6} + 2 x^{4} - x^{2} - x + 1$.

Definition

A Salem number is an algebraic integer $\tau>1$ whose other conjugates have absolute value at most $1$, with at least one on the unit circle [5]. Then $\tau^{-1}$ is a conjugate and the rest lie on the unit circle. Listed are the small Salem numbers, those less than $1.3$.

Parameters

coefficients

— leading coefficients of the minimal polynomial (the coefficients $a_0,\ldots,a_{d/2}$ of $x^d,\ldots,x^{d/2}$ in the minimal polynomial of a Salem number of even degree $d$)

Formulas

(1)

For coefficients $a_0,a_1,\ldots,a_{d/2}$, the minimal polynomial is $a_0x^d+a_1x^{d-1}+\cdots+a_{d/2}x^{d/2}+\cdots+a_1x+a_0$.

Comments

(2)

The Mahler measure of a Salem number's minimal polynomial is $\tau$, so its logarithmic Mahler measure is $\log \tau$. Lehmer's number, the first entry here, is the smallest known Mahler measure greater than $1$ among non-cyclotomic integer polynomials.

(3)

The bound $1.3$ is traditional rather than only a size cutoff: Salem numbers below $1.3$ are called small in the literature [1]. The plastic constant, the real root of $x^3-x-1$, is the smallest known limit point of the Salem numbers; a construction due to R. Salem gives infinitely many Salem numbers below it, and $1.3$ lies below that accumulation point [1].

Programs

(P1)

Sage

R.<x> = ZZ[]
half = [1, 1, 0, -1, -1, -1]
d = 2*(len(half) - 1)
f = sum(half[i]*x^(d-i) for i in range(len(half))) + sum(half[i]*x^i for i in range(len(half)-1))
max(f.roots(RealIntervalField(400), multiplicities=False))

References

[1]

J.-M. Sac-Épée, Salem numbers less than $49/37$, 2025. (arXiv) (doi)

[2]

W. J. Floyd, Growth of planar Coxeter groups, P.V. numbers, and Salem numbers, Mathematische Annalen 293 (1992), no. 3, 475-483. (zbMATH) (MR)

[3]

M. J. Mossinghoff, G. Rhin, and Q. Wu, Minimal Mahler measures, Experiment. Math. 17 (2008), no. 4, 451-458. (doi) (MR)

[4]

M. J. Mossinghoff, Polynomials with small Mahler measure, Math. Comp. 67 (1998), no. 224, 1697-1705, S11-S14. (doi)

Links

[5]

Wikipedia: Salem number

[6]

Mossinghoff: Small Salem Numbers

Similar tables

Pisot numbers less than the golden ratio — its first entry is the plastic constant, the smallest known limit point of the Salem numbers and the accumulation point that bounds the traditional small-Salem range

Growth rates of hyperbolic Coxeter triangle groups — its growth rates for $\Delta(2,3,r)$ with $7\leq r\leq11$ and for $\Delta(2,4,5)$ are Salem numbers held here; finite-label hyperbolic Coxeter triangles have Salem growth rates [2]

Growth rates of hyperbolic Coxeter simplex groups — growth rates of the compact hyperbolic Coxeter tetrahedra, all Salem numbers and all greater than $1.3$

Mahler measures of $x+x^{-1}+y+y^{-1}+k$ — holds logarithmic Mahler measures for Boyd's two-variable family rather than Mahler measures of Salem minimal polynomials

Mahler measures of $1+x_1+\dots+x_{n-1}$ — holds logarithmic Mahler measures for short random walks rather than Mahler measures of Salem minimal polynomials

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the 47 known Salem numbers below $1.3$ in Mossinghoff's list [6]. The list is proved complete for degree at most $44$ [3]. The one degree-$46$ entry lies outside that range, and a later random-sampling search through Salem degree $64$ rediscovered all 47 and found no others below $1.3$ [1])

How they were obtained:

The generator rebuilds each reciprocal polynomial from the coefficient parameter, checks that it is irreducible and has the Salem root pattern, isolates its real root greater than $1$ in interval arithmetic, and writes $100$ digits. Mossinghoff's source decimals are used only as a truncation check.