Constants

Ramsey numbers

Cmglee, via Wikimedia Commons. CC BY-SA 4.0.

Ramsey number $R(r,s)$ is the smallest number $n$ such that any two-coloring of the edges of the complete graph $K_n$ must contain either a monochromatic $K_r$ in the first color or a monochromatic $K_s$ in the second color.

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r\s	3	4	5	6	7	8	9	10	11	12	13	14	15
3	$6$	$9$	$14$	$18$	$23$	$28$	$36$	$40$ $41$	$47$ $50$	$53$ $59$	$60$ $68$	$67$ $77$	$74$ $87$
4		$18$	$25$	$36$ $40$	$49$ $58$	$59$ $79$	$73$ $105$	$92$ $135$	$102$ $170$	$128$ $210$	$138$ $256$	$147$ $307$	$158$ $364$
5			$43$ $46$	$59$ $85$	$80$ $133$	$101$ $193$	$133$ $282$	$149$ $381$	$183$ $511$	$203$ $672$	$233$ $860$	$267$ $1081$	$275$ $1341$
6				$102$ $160$	$115$ $270$	$134$ $423$	$183$ $651$	$204$ $944$	$262$ $1346$	$294$ $1855$	$347$ $2499$	$?$ $3301$	$401$ $4305$
7					$205$ $492$	$219$ $832$	$252$ $1368$	$292$ $2119$	$405$ $3197$	$417$ $4665$	$511$ $6653$	$?$ $9260$	$?$ $12635$
8						$282$ $1518$	$329$ $2662$	$343$ $4402$	$457$ $7040$	$?$ $10836$	$817$ $27485$	$?$ $41525$	$873$ $63609$
9							$565$ $4956$	$581$ $8675$	$?$ $14631$	$?$ $38832$	$?$ $64864$	$?$ $?$	$?$ $?$
10								$798$ $16064$	$?$ $45881$	$?$ $81123$	$?$ $?$	$?$ $?$	$1313$ $?$

Packing unit squares in a square

Optimal packing of 5 unit squares in a square

Amit6, via Wikimedia Commons. Public domain.

$s(n)$ is the side of the smallest square into which $n$ unit squares can be packed.

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$1$

$2$

$2 + \frac{1}{\sqrt{2}}$

$3$

$3 + \frac{1}{\sqrt{2}}$

$2 + \frac{4}{\sqrt{5}}$
${\approx}3.8771$

$2 + \frac{4}{\sqrt{5}}$
$4$

$3.8437$
$4$

$4$

$\frac{40\sqrt{2}+19}{17}$
${\approx}4.6756$

$\frac{40\sqrt{2}+19}{17}$
$\frac{7+\sqrt{7}}{2}$

$6\sqrt{2}-4$
$3+\frac{4\sqrt{2}}{3}$

$2\sqrt{2}+\frac{27+2\sqrt{10}}{13}$
$\frac{7}{2}+\frac{3\sqrt{2}}{2}$

$2\sqrt{2}+\frac{27+2\sqrt{10}}{13}$
$5+\frac{1}{\sqrt{2}}$

$2\sqrt{2}+\frac{6}{\sqrt{5}}$
$3+2\sqrt{2}$

$2\sqrt{2}+\frac{6}{\sqrt{5}}$
${\approx}5.9344$

$2\sqrt{2}+\frac{6}{\sqrt{5}}$
$6$

$5.6415$
$6$

$?$
$6$

$6$

$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
${\approx}6.5987$

$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
$6+\frac{1}{\sqrt{2}}$

$2\sqrt{2}+\frac{113+10\sqrt{3}}{37}$
${\approx}6.8189$

$2\sqrt{2}+\frac{8}{\sqrt{5}}$
$4+2\sqrt{2}$

$2\sqrt{2}+\frac{8}{\sqrt{5}}$
${\approx}6.9473$

$?$
$7$

$7$

$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
${\approx}7.5987$

$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
${\approx}7.7044$

$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
$7+\frac{1}{\sqrt{2}}$

$2\sqrt{2}+\frac{101+3\sqrt{14}}{25}$
${\approx}7.8231$

$?$
${\approx}7.8488$

$?$
${\approx}7.9871$

$?$
$8$

$8$

$2\sqrt{2}+\frac{71}{13}$
$5+\frac{5}{\sqrt{2}}$

$2\sqrt{2}+\frac{71}{13}$
$3+4\sqrt{2}$

$2\sqrt{2}+\frac{71}{13}$
$8+\frac{1}{\sqrt{2}}$

$2\sqrt{2}+\frac{71}{13}$
$\frac{15}{2}+\frac{\sqrt{7}}{2}$

$?$
${\approx}8.8562$

$?$
${\approx}8.9121$

$?$
${\approx}8.9633$

$?$
$9$

$9$

${\approx}9.2667$
$6+\frac{5}{\sqrt{2}}$

${\approx}9.2667$
$4+4\sqrt{2}$

${\approx}9.2667$
$9+\frac{1}{\sqrt{2}}$

${\approx}9.2667$
$\frac{11}{2}+3\sqrt{2}$

$?$
$\frac{17+\sqrt{7}}{2}$

$?$
${\approx}9.8520$

$?$
${\approx}9.9018$

$?$
$5+\frac{7}{\sqrt{2}}$

$?$
$10$

$10$

100

$10$

Busy Beaver shift function

Simplified spacetime diagram of the fifth busy beaver machine

--MULLIGANACEOUS--, via Wikimedia Commons. Public domain.

The Busy Beaver shift function $BB(n)$ ($S(n)$) is the maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting.

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$7.4\times10^{36524}$
$?$

$10^{2\times10^{10^{10^{18705353}}}}$
$?$

Busy Beaver ones function

OrdinaryArtery, via Wikimedia Commons. CC BY-SA 4.0.

The Busy Beaver ones function $\Sigma(n)$ (Rado’s sigma function) is the maximal number of 1s that an n-state Turing machine can print on an initially blank tape before eventually halting.

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$3.5\times10^{18267}$
$?$

$10^{10^{10^{10^{18705353}}}}$
$?$

Kissing numbers

Robertwb at English Wikipedia, via Wikimedia Commons. CC BY-SA 3.0.

The kissing number of $\mathbb{R}^n$ is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^n$.

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Degree-diameter problem

$N(d,k)$ is the largest possible number of vertices in a graph of maximum degree $d$ and diameter $k$.

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d\k	2	3	4	5	6	7	8	9	10
3	$10$	$20$	$38$	$70$ $?$	$132$ $?$	$196$ $?$	$360$ $?$	$600$ $?$	$1250$ $?$
4	$15$	$41$ $?$	$98$ $?$	$364$ $?$	$740$ $?$	$1320$ $?$	$3243$ $?$	$7575$ $?$	$17703$ $?$
5	$24$	$72$ $?$	$212$ $?$	$624$ $?$	$2772$ $?$	$5516$ $?$	$17030$ $?$	$57840$ $?$	$187056$ $?$
6	$32$	$111$ $?$	$390$ $?$	$1404$ $?$	$7917$ $?$	$19383$ $?$	$76891$ $?$	$331387$ $?$	$1253615$ $?$
7	$50$	$168$ $?$	$672$ $?$	$2756$ $?$	$12264$ $?$	$53020$ $?$	$249660$ $?$	$1223050$ $?$	$6007230$ $?$
8	$57$ $?$	$253$ $?$	$1100$ $?$	$5115$ $?$	$39672$ $?$	$131137$ $?$	$734820$ $?$	$4243100$ $?$	$24897161$ $?$
9	$74$ $?$	$585$ $?$	$1640$ $?$	$8268$ $?$	$75893$ $?$	$279616$ $?$	$1697688$ $?$	$12123288$ $?$	$65866350$ $?$
10	$91$ $?$	$650$ $?$	$2331$ $?$	$13203$ $?$	$134690$ $?$	$583083$ $?$	$4293452$ $?$	$27997191$ $?$	$201038922$ $?$
11	$104$ $?$	$715$ $?$	$3200$ $?$	$19620$ $?$	$156864$ $?$	$1001268$ $?$	$7442328$ $?$	$72933102$ $?$	$600380000$ $?$
12	$133$ $?$	$786$ $?$	$4680$ $?$	$29621$ $?$	$359772$ $?$	$1999500$ $?$	$15924326$ $?$	$158158875$ $?$	$1506252500$ $?$
13	$162$ $?$	$856$ $?$	$6560$ $?$	$40488$ $?$	$531440$ $?$	$3322080$ $?$	$29927790$ $?$	$249155760$ $?$	$3077200700$ $?$
14	$183$ $?$	$916$ $?$	$8200$ $?$	$58095$ $?$	$816294$ $?$	$6200460$ $?$	$55913932$ $?$	$600123780$ $?$	$7041746081$ $?$
15	$187$ $?$	$1215$ $?$	$11712$ $?$	$77520$ $?$	$1417248$ $?$	$8599986$ $?$	$90001236$ $?$	$1171998164$ $?$	$10012349898$ $?$
16	$200$ $?$	$1600$ $?$	$14640$ $?$	$132496$ $?$	$1771560$ $?$	$14882658$ $?$	$140559416$ $?$	$2025125476$ $?$	$12951451931$ $?$
17	$274$ $?$	$1610$ $?$	$19040$ $?$	$133144$ $?$	$3217872$ $?$	$18495162$ $?$	$220990700$ $?$	$3372648954$ $?$	$15317070720$ $?$
18	$274$ $?$	$1620$ $?$	$23800$ $?$	$171828$ $?$	$4022340$ $?$	$26515120$ $?$	$323037476$ $?$	$5768971167$ $?$	$16659077632$ $?$
19	$338$ $?$	$1638$ $?$	$23970$ $?$	$221676$ $?$	$4024707$ $?$	$39123116$ $?$	$501001000$ $?$	$8855580344$ $?$	$18155097232$ $?$
20	$381$ $?$	$1958$ $?$	$34952$ $?$	$281820$ $?$	$8947848$ $?$	$55625185$ $?$	$762374779$ $?$	$12951451931$ $?$	$78186295824$ $?$

Cages

David Benbennick, via Wikimedia Commons. Public domain.

$n(k,g)$ is the order of a $(k,g)$-cage, a $k$-regular graph of girth $g$ of minimum order.

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k\g	3	4	5	6	7	8	9	10	11	12	13	14	15	16
3	$4$	$6$	$10$	$14$	$24$	$30$	$58$	$70$	$112$	$126$	$202$ $272$	$258$ $384$	$384$ $620$	$512$ $960$
4	$5$	$8$	$19$	$26$	$67$	$80$	$?$ $275$	$?$ $384$	$?$ $?$	$728$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
5	$6$	$10$	$30$	$42$	$?$ $152$	$170$	$?$ $?$	$?$ $1296$	$?$ $2688$	$2730$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
6	$7$	$12$	$40$	$62$	$?$ $294$	$312$	$?$ $?$	$?$ $?$	$?$ $?$	$7812$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
7	$8$	$14$	$50$	$90$	$?$ $?$	$?$ $672$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $32928$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
8	$?$ $?$	$?$ $?$	$67$ $80$	$114$	$?$ $?$	$?$ $800$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $39216$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
9	$?$ $?$	$?$ $?$	$86$ $96$	$146$	$?$ $1152$	$?$ $1170$	$?$ $?$	$?$ $?$	$?$ $74752$	$?$ $74898$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
10	$?$ $?$	$?$ $?$	$103$ $124$	$182$	$?$ $?$	$?$ $1640$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $132860$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
11	$?$ $?$	$?$ $?$	$124$ $154$	$224$ $240$	$?$ $?$	$?$ $2618$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $319440$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
12	$?$ $?$	$?$ $?$	$147$ $203$	$266$	$?$ $?$	$?$ $2928$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $354312$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
13	$?$ $?$	$?$ $?$	$174$ $230$	$314$ $336$	$?$ $?$	$?$ $4342$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $738192$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
14	$?$ $?$	$?$ $?$	$199$ $288$	$366$	$?$ $?$	$?$ $4760$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $804468$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
15	$?$ $?$	$?$ $?$	$230$ $312$	$422$ $462$	$?$ $?$	$?$ $7648$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $1957376$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
16	$?$ $?$	$?$ $?$	$259$ $336$	$482$ $504$	$?$ $?$	$?$ $8092$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $2088960$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
17	$?$ $?$	$?$ $?$	$294$ $448$	$546$	$?$ $?$	$?$ $8738$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $2236962$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
18	$?$ $?$	$?$ $?$	$327$ $480$	$614$	$?$ $?$	$?$ $10440$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $3017196$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
19	$?$ $?$	$?$ $?$	$364$ $512$	$686$ $720$	$?$ $?$	$?$ $13642$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $4938480$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
20	$?$ $?$	$?$ $?$	$403$ $576$	$762$	$?$ $?$	$?$ $14480$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $5227320$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$

de Bruijn-Newman constant

The de Bruijn-Newman constant $\Lambda$ is defined via the zeros of a certain function $H(\lambda,z)$, where $\lambda$ is a real parameter and $z$ is a complex variable. More precisely, $H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) \, du$, where $\Phi$ is the super-exponentially decaying function $\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}$ and $\Lambda$ is the unique real number with the property that $H$ has only real zeros if and only if $\lambda\geq \Lambda$.

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$\ge 0$
$\le 0.2$

Van der Waerden numbers

$W(r,k)$ is the smallest number $n$ such that if the integers $1$ to $n$ are colored with $r$ colors, there must be a monochromatic arithmetic progression of length $k$.

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r\k	3	4	5	6	7	8	9	10	11
2	$9$	$35$	$178$	$1132$	$3704$ $?$	$11496$ $?$	$41266$ $?$	$103475$ $?$	$193942$ $?$
3	$27$	$293$	$2174$ $?$	$11192$ $?$	$48812$ $?$	$238401$ $?$	$932746$ $?$	$4173725$ $?$	$18603732$ $?$
4	$76$	$1049$ $?$	$17706$ $?$	$157210$ $?$	$2284752$ $?$	$12288156$ $?$	$139847086$ $?$	$1189640579$ $?$	$3464368084$ $?$
5	$171$ $?$	$2255$ $?$	$98742$ $?$	$786741$ $?$	$15993258$ $?$	$86017086$ $?$	$978929596$ $?$	$8327484047$ $?$	$38108048914$ $?$
6	$226$ $?$	$9779$ $?$	$98749$ $?$	$1555550$ $?$	$111952800$ $?$	$602119596$ $?$	$6852507166$ $?$	$58292388323$ $?$	$419188538044$ $?$

Stamp folding

Robert Dickau, via Wikimedia Commons. CC BY-SA 3.0.

The number of distinct ways to fold a strip of $n$ labeled stamps.

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Map folding

Robert Dickau, via Wikimedia Commons. CC BY-SA 3.0.

The number of distinct ways to fold an $m \times n$ rectangle into a unit square.

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m\n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
2	$8$	$60$	$320$	$1980$	$10512$	$60788$	$320896$	$1787904$	$9381840$	$51081844$	$266680992$	$1429703548$	$7432424160$	$39409195740$	$204150606976$	$1073644675448$	$5545305620064$
3		$1368$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
4			$300608$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
5				$186086600$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
6					$123912532224$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$
7						$129950723279272$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$	$?$ $?$