rcs>Was this checked for all N & Q where N+2Q < 458M?
The nominal work is ~10^17. I can see some algorithm possibilities to improve this, but I'm curious about what's actually been done.
Rich ----- Yes! It's actually almost linear in q. Let m:=458M. For q=1,2,... you scan backward for a q s.t. a[m]-2*a[m-q]+a[m-2q] = 0. Then you scan forward k=1,2,... for 2*a[m-q+k]-a[m-2q+k]<=0, which is usually pretty prompt. --Bill ---------- Quoting Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>>:
Denote by a[0], a[1], ... the continued fraction terms of pi> (= 3,7,15,1,... = A001203). If there are rationals A,B,C,D such that> pi = (Ae+B)/(Ce+D), then there is a quasiperiod q and a preamble p> such that the second differences>> a[n+2q]-2a[n+q]+a[n] = 0>> for all n>p. For e itself, q=3 and p=1. (Or any positive multiple of 3> and any positive integer). E.g., define a second-differencer:> In[13]:= dif2[L_List, q_Integer:1] :=> Drop[L, 2*q] - 2*Drop[Drop[L, -q], q] + Drop[L, -2*q]>> In[16]:= dif2[{a, b, c, d, e}, 2]>> Out[16]= {a - 2 c + e}> Then> In[163]:= dif2[ContinuedFraction[E, 22], 3]>> Out[163]= {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}> (p=1,q=3).> In[166]:= dif2[ContinuedFraction[E^(2/3), 22], 5]>>> Out[166]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}>> (p=0,q=5).>> In[171]:= dif2[ContinuedFraction[(E - 1)/(1 - E/6), 333], 130]>>>> Out[171]= {2, 1, 11, -1, 0, 62, 1, 3, 0, -1, 0, 1, -5, 5, 5, 1, -2, \>> -153, -2, 0, 1, -58, 5, 2, 1, 152, 2, 57, -6, 3, -2, -21, 6, -10, 3, \>> 22, -18, 3, 18, 4, 0, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \>> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}>> (p=44, q=130)>>>> By second-differencing Neil's 458585379 terms one can contrive at most>> seven(!)>> terminal 0s, with, e.g., p=458585379-2q, q=16400301. But not even these>> seven>> can signal an incipient quasiperiod, because annihilating all future second>> differences requires that some yet-to-be-computed terms be nonpositive! A>> program has confirmed that *all* terminal bursts, even of length one, are>> similarly doomed, thereby precluding any e-like (Hurwitz) behavior with>> p+2q < 458585379. Thus at least one of A,B,C,D has a big numerator or>> denominator. (How big?)>> --rwg>> Maybe not huge. For 3.141595... = 4-6e/19, p=447 and q=676.>>