The program I wrote has stumbled upon one of the messiest sets of equations yet.
It relates to two noticeable correlations between constants. The first is between the Twin Prime Reciprocal Average (TPRA) and the Lower Iterated Exponential Function (LIEF). I substituted in the value of a known constant -- the Reuleaux Tetrahedron Volume -- and solved for the TPRA, to arrive at it.
The second is between the Lower Iterated Exponential Function and two other constants: the smallest known Salem Number and the Landau-Ramanujan constant.
I explained in a previous post what a Twin Prime was: a pair of back-to-back prime numbers, each of which is divisible only by itself and 1. When you average the value of a Twin Prime (e.g. (5+7)/2 = 6) and then take its reciprocal (6 becomes 1/6), summing all such numbers together approaches the TPRA constant.
The Lower Limit of the Iterated Exponential Function pertains to the infinite tetration from n=2 to infinity of 1/n, otherwise known as the "Power Tower". Raise 1/9 to the 1/10th power, then raise 1/8 to the result, and raise 1/7 to the next result, until you reach 1/2. The resulting number is bounded between around 0.658 at the lower end and 0.69 at the upper end.
I will hopefully go over the other constants in subsequent posts, but the important part is that they are known to many more digits than the LIEF Constant, which allowed me to make a guess at its value that falls within its known error bounds:
LIEF ~= asinh(ln(ln(Salem)) / (ln(LR) - ln(5) - ln(2)))
{Salem = 1.17628081825991750654407033847403505; LR = 0.76422365358922066299069873125009232}
Or:
H2n ~= sinh-1(log(ln(K)-ln(5)-ln(2))(ln(σ10)))
{K = Landau-Ramanujan; σ10 = Salem}
H2n ~= 0.6583655997621672859686154218045069683492221974225818865
The second equation, attempting to guess the Twin Prime Reciprocal Average, is:
TPRA ~= sqrt((10^((49*Pi)/(12*LIEF) - 1/(2*sqrt(2)*LIEF) - (27*atan(sqrt(2)))/(2*LIEF)))^(1/10))
{Pi = 3.14159265358979323846264338327950288; LIEF = 0.6583655997621672859686154218045069683492221974225818865}
In mathematical notation, the equation is:
B1 ~= (10[(49*π)/(12*H2n)-1/(2*2(1/2)*H2n)-(27*tan-1(2(1/2)))/(2*H2n)])(1/10)(1/2)
{π = Pi; H2n = Lower Iterated Exponential Function}
B1 ~= 0.928835827233491767916238344645865964701808
Just a word of note: the above formula might be messy owing to the fact that there is no known equation for all but one of the constants used to calculate it. I would be interested in substituting any such equations, should they ever be discovered, and seeing whether the result can be simplified to something much more concise. In the meantime, the above equation is the closest that my computer program has come to expressing the Twin Prime Reciprocal Average in terms of other more familiar mathematical constants.
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