# Conway’s constant

Conway’s constant $\lambda\approx 1.303577296$ gives the asymptotic rate of growth in the length between $a_{i}$ and $a_{i+1}$ in most look and say sequences. That is, given a function $d(n)$ that gives us the number of digits of $n$ in base 10, then

 $\lim_{i\to\infty}\frac{a_{i+1}}{a_{i}}=\lambda.$

For example, starting with $n=1$ and skipping ahead to $a_{7}$, we observe

$i$ $a_{i}$ $\frac{a_{i+1}}{a_{i}}$
7 13112221 1.333333333…
8 1113213211 1.25
9 31131211131221 1.4
10 13211311123113112211 1.428571428…
11 11131221133112132113212221 1.3
12 3113112221232112111312211312113211 1.307692307…

Conway’s constant is the largest zero of this degree 71 polynomial:

$x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60% }-x^{59}$ $+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}+x^{4% 9}+9x^{48}$ $-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}-12x^{39}+7x^% {38}-7x^{37}$ $+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}-10x^{29}-3x^{28}+2x^{2% 7}+9x^{26}$ $-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}+3x^{19}-4x^{18}-10x^{17}-7x^{16}+12x% ^{15}$ $+7x^{14}+2x^{13}-12x^{12}-4x^{11}-2x^{10}+5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12% x^{3}-6x^{2}+3x-6$

## References

• 1 Steven R. Finch, Mathematical Constants. Cambridge: Cambridge University Press (2003): 453
Title Conway’s constant ConwaysConstant 2013-03-22 18:02:36 2013-03-22 18:02:36 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A63