Someone helped me see a couple errors in the previous: "the series could converge but the partial sums diverge" was wrong (a mis-use of words). I should have said "the series could diverge and yet have occasional partial sums that are as close as desired" or something like that. Also, at the beginning I had "s=640320^3" (to match Gosper), then later (in the bc commands) I re-defined "s=640320". It might have been better if I used "k=640320". The below quote is altered appropriately, if anyone cares (-: - Robert On Wed, Dec 22, 2010 at 19:39, Robert Munafo <mrob27@gmail.com> wrote:

I wasn't able to find anything definitive, but I'll add that one link I did find.

Most of the related work looks at the more universal properties of such numbers, related to the j-function and other deep mathematical truths, which is beyond my ability (or at least my patience :-)

But it is fairly easy to see that beyond the point where A178449 and A178451 start to differ, there is a lot of wiggle-room for the coefficients.

Suppose it's just a numerical approximation we're after. Consider the basic identity of this particular series sum:

e^(pi sqrt(163)) = s + C_0 - C_1/s + C_2/s^2 - C_3/s^3 + C_4/s^4 - C_5/s^5 + C_6/s^6 - ...

(where s=640320^3 = 262537412640768000), one could use a multi-precision calculator and find non-negative integers C_n such that each partial sum is as close as possible to the limit, then clearly no C_n need be larger than s.

However, individual C_n could be "non-optimal", and the departure from convergence made up in later terms. Or the series could diverge and yet have occasional partial sums that are as close as desired, for any number of arcane reasons.

In this particular conjecture, the terms do stay in the range 0<=C_n<=s, and the partial sums also seem to converge as quickly as one would hope. Here I have used 200 digits but truncate the output to show just 185 digits beyond the decimal point (using bc, I can supply functions if you're interested):

setscale(200) Setting scale to 200 Computing: phi = 1.618033... e = 2.718281... pi = 3.141592... 0 pi

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948

k=640320

k^3 262537412640768000

1/k^18

.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000305388581546670798710822103120316933232323133422616537384593020687404558608343755

Here I have inserted a "_" between the last digit of agreement and the first digit that differs:

exp(pi * sqrt(163))

262537412640768743.99999999999925007259719818568887935385633733699086270753741037821064791011860731295118134618606450419308_388794975386404490572871447719681485232243203911647829148864228272013117831705905

t = k^3 + 744 - 196884/k^3 + 167975456/k^6 - 180592706130/k^9 + 217940004309743/k^12 - 19517553165954887/k^15 + 74085136650518742/k^18 t

262537412640768743.99999999999925007259719818568887935385633733699086270753741037821064791011860731295118134618606450419308_541556967018229027734370849526479008227498474707036235656894699095863661999704040

and now showing that the coefficient C_6 is indeed as close as it can be:

(t - exp(pi * sqrt(163))) * k^18 .50022168758945164251081982667278829322179...

One could simply use a numerical technique to continue the series in the same way, giving (for example) 131326907606533204 for C_7 -- but I don't know the intention of Gosper (or the original author behind his source) ...

- Robert

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