RCS> 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 --------- I lose; you win: In[11]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 12] Out[11]= 3366937412175501591717367777101460820640224690509113100163277\ 6539067634240985481258893779792671452248127456725155260727280357834657\ 6076738692420210769091266874710826881662611178920169012031118635505607\ 4310982786125119105965083788824019629431580952133240830246962982908185\ 9746496798634116562354568928408751636377986351524509214103104546223261\ 2396128362132130189597314029272968730759961945103880790832377276137373\ 9360515945586462465327970571098621823207535551580401784234812938313843\ 6459926693612089970288170337138209267562435148320741084111322614415098\ 1287168301351535936792810718614137830956322091599451719533327897728719\ 2022822793035110202282332960887794400493112644185559756058506634629406\ 8091290492788783650041163770989366969890285626688901808066658454549560\ 7443533211603079106672889774858913012113951640476740718704380411099010\ 204 In[13]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 6] Out[13]= 4457343787994 (Note: a_0:=2 .) In[14]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[14]= 2 In[16]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 4] Out[16]= 1103 In[17]:= PowerMod[3, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[17]= 3 In[18]:= PowerMod[8, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[18]= 8 In[12]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 13]]] Out[12]= {0.105467, 2.415419924381151*10^1667} In[19]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 14]]] Out[19]= {0.647739, 7.344539857481122*10^3334} In[20]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 15]]] Out[20]= {3.835341, 6.597333832436021*10^6669} In[21]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 16]]] Out[21]= {21.430643, 1.412908137918137*10^13339} In[22]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 17]]] Out[22]= {119.182441, 8.856579016766857*10^26679} In[23]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 18]]] Out[23]= {576.777472, 7.425006686698617*10^53359} In[24]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 18]]] Out[24]= {577.039477, False} In[25]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 19]]] should take about an hour. Meanwhile, we can now conjecture, as with Fermat #s, that there are no more of these primes! --rwg 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 SUPERIMPOSED PSEUDOPRIMES <disformatting earlier comments --Firefox>