I'll guess that the False value for a_20 is from some small factor, and that the Aborted values are because the test ran out of time before completing even one superimposed test. Try asking for a few digits of 2^(a_12) (mod a_12), and then see how far you can get with a_13 ... before running out of gas. Rich --------- Quoting Bill Gosper <billgosper@gmail.com>: <eliding earlier comments --rich>
Only slightly less infeasible might be the coprime sequence A000058 (interesting comments). In[10]:= NestList[#^2 - # + 1 &, 2, 9]
Out[10]= {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443}
In[8]:= N[Timing[Nest[#^2 - # + 1 &, 2, 21]]]
Out[8]= {0.007302, 1.853733657356281*10^426880}
(Half a million digits in 7ms?)
In[6]:= TimeConstrained[PrimeQ[#], 1] & /@ NestList[#^2 - # + 1 &, 2, 21]
Out[6]= {True, True, True, True, False, True, False, False, False, False, False, False, False, False, False, False, False, False, $Aborted, $Aborted, False, $Aborted}
where $Aborted almost certainly means True. Changing the timeout from 1 to 288 sec did zilch. Decertifying a_20 In[9]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 20]]]
Out[9]= {0.150525, False}
took < 1/6 sec. Can we show there are no (strong?) pseudoprimes of this form?
--rwg
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