The Conway Sequence, or the "Look and Say" sequence, is based on a simple word game. Start with the number 1, and then speak the number out loud: "There is one 1". You've just created the next number in the sequence: 11. Now say that numerical sequence aloud: "There are two 1s". the next number is therefore 21. Keep going, and the number grows at a rate which at first is logarithmic, but then levels out to a straight(ish) line, with a slope that hovers around a single number: Conway's Constant.
Conway was able to calculate and then solve a very complicated 71st-order polynomial function which had the constant as its solution. Why did he do this? For the same reason I'm now investigating his constant: because he found it fun to play around with numbers.
The program I wrote found a correlation almost as bizarre as the original number itself. When substituting the known values for the other constants into the equation -- ii and the Madelung Constant for measuring electrostatic potential in Ions -- the result was the following equation in terms of e and π:
Conway's Constant ~= tanh(e^(-Pi/2))^(-Pi*ln(3)*sqrt(3))/((10^2)^2)
{Pi = 3.141592653589793238462643383279502884197169; e = 2.718281828459045235360287}
In mathematical notation:
λ ~= tanh(e(-π/2))-π*ln(3)*3(1/2)/(102)2
{π = Pi}
λ ~= 1.30357727444243995581509783875403
As with the other numerical correlations, I have no real explanation for why these two numbers are so similar, and computing power is as of yet not appreciable enough to either verify or disprove this as well as other equations. Perhaps it one day will be though, because the number-crunching demands of proofs such as these far exceed the human mind's abilities, so if they are forever outside the realm of computational ability as well, then they quite simply cannot be done.
Add new comment