sequence (x+1)^x + x^x and Not too bad !!

[Farideh Firoozbakht]

Hello ,
 
  
   " 3,13,881 and next prime has more than 147962 digits. "
 
 
If f(x)=(x+1)^x+x^x is prime then x must be of the form 2^m.
Because if x has odd prime factor p, x=p*s ,then
 (x+1)^x+x^x=((x+1)^s)^p+(x^s)^p and can not be prime.
 
x=2^0  f(x)=3
x=2^1  f(x)=13
x=2^2  f(x)=881
 
If x=2^m for m=3,...,4 f(x) is composite,thus next prime occurs for 
x > 32767 and has more than 147962 digits.
 
" In[38]:=
  Do[v = (2^m + 1)^(2^m) + (2^m)^(2^m); 
  Print[Timing[{m, PrimeQ[v]}]], {m, 0, 20}]
 
  From In[38]:=
  {0. Second, {0, True}}
 
  From In[38]:=
  {0. Second, {1, True}}
 
  From In[38]:=
  {0. Second, {2, True}}
 
  From In[38]:=
  {0. Second, {3, False}}
 
  From In[38]:=
  {0. Second, {4, False}}
 
  From In[38]:=
  {0. Second, {5, False}}
 
  From In[38]:=
  {0. Second, {6, False}}
 
  From In[38]:=
  {0.063 Second, {7, False}}
 
  From In[38]:=
  {0.375 Second, {8, False}}
 
  From In[38]:=
  {3.453 Second, {9, False}}
 
  From In[38]:=
  {26.078 Second, {10, False}}
 
  From In[38]:=
  {206.172 Second, {11, False}}
 
  From In[38]:=
  {1130.53 Second, {12, False}}
 
  From In[38]:=
  {8818. Second, {13, False}}
 
  From In[38]:=
  {40623.6 Second, {14, False}}
 
  Out[38]=
  $Aborted
 
  In[46]:=
{2^15 - 1, Length[IntegerDigits[(2^15 + 1)^(2^15) + (2^15)^(2^15)]] - 1}
 
  Out[46]=
  {32767, 147962} "
 
 
Best wishes,
Farideh.
 
 
 
Quoting "Pfoertner, Hugo" <Hugo.Pfoertner at muc.mtu.de>:
 
> -----Original Message-----
> 
> Cino Hilliard's message contained (in part):
> > >      print1(3" ");
> > >      forstep(x=2,n,2,
> > >        y=(x+1)^x+x^x;
> > >         if(isprime(y),print1(y" "))
> > >
> > > Maybe someone can further limit the search or find more terms?
> >
> 
> To which Jim Nastos replied (in part):
> >   Since primality testing is costly for large numbers,
> [...]
> 
> R. Shepherd replied:
> 
> > To which I add/clarify that it's my experience that even Pari's
> *probable*-
> > primality testing is costly for sufficiently large numbers
> [...]
> 
> I have now tested up to x=6000, for which (x+1)^x+x^x is ~ 3*10^22669
> and found nothing. Too bad, but I think it's time to give up here :-(
> 
> Hugo
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20030831/f70be70b/attachment.htm>