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%I #22 Aug 29 2014 14:04:37
%S 7,17,79,431,58049,130783,162287,523927,2486784401,6102977801,
%T 8375575711,13055867207,83695120256591,375700268413577,
%U 2251799813682647,9007199254738183,79792265017612001,1490116119372884249
%N Primes of the form x^y - y^x, for x,y > 1.
%C These are the primes in A045575, numbers of the form x^y - y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k - k^n) for k>1.
%C If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)-y^x. - _M. F. Hasler_, Aug 19 2014
%H T. D. Noe, <a href="/A123206/b123206.txt">Table of n, a(n) for n=1..101</a> (terms < 10^400)
%H H. Lifchitz & R. Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=x%5Ey-y%5Ex&action=Search">PRP of the form x^y-y^x</a> on primenumbers.net.
%e The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014
%p N:= 10^100: # to get all terms <= N
%p A:= NULL:
%p for x from 2 while x^(x+1) - (x+1)^x <= N do
%p for y from x+1 do
%p z:= x^y - y^x;
%p if z > N then break
%p elif z > 0 and isprime(z) then A:=A, z;
%p fi
%p od od:
%p {A}; # _Robert Israel_, Aug 29 2014
%t Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
%t nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num<nn, Sow[num]; While[k++; num=n^k-k^n; num<nn, Sow[num]]]][[2,1]]]; Select[t, PrimeQ]
%t With[{nn=30},Take[Sort[Select[#[[1]]^#[[2]]-#[[2]]^#[[1]]&/@Subsets[ Range[ 2nn],{2}],#>0&&PrimeQ[#]&]],nn]] (* _Harvey P. Dale_, Nov 23 2013 *)
%o (PARI) a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - _M. F. Hasler_, Aug 19 2014
%Y Cf. A045575, A122735, A078202, A082754, A055651, A094133.
%Y A163319 is the subsequences for fixed x=3, A243114 for x=6.
%Y Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Oct 04 2006
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