A045575 - OEIS
A045575
Nonnegative numbers of the form x^y - y^x, for x,y > 1.
5
0, 1, 7, 17, 28, 79, 118, 192, 399, 431, 513, 924, 1844, 1927, 2800, 3952, 6049, 7849, 8023, 13983, 16188, 18954, 32543, 58049, 61318, 61440, 65280, 130783, 162287, 175816, 255583, 261820, 357857, 523927, 529713, 1038576, 1048176
COMMENTS
Pillai proved that there are ~ 0.5 * (log x)^2/(log log x)^2 terms of this sequence up to x. - Charles R Greathouse IV, Jul 20 2017
Conjecture: For d > 11, 10^d - d^10 is the largest (base-ten) d-digit term. - Hans Havermann, Jun 12 2023
REFERENCES
S. S. Pillai, On the indeterminate equation x^y - y^x = a, Journal Annamalai University 1, Nr. 1, (1932), pp. 59-61. Cited in Waldschmidt 2009.
MAPLE
N:= 10^8: # to get all terms <= N
A:= (0, 1):
for x from 2 while x^(x+1) - (x+1)^x <= N do
for y from x+1 do
z:= x^y - y^x;
if z > N then break
elif z > 0 then A:=A, z;
fi
od od:
MATHEMATICA
Union[Flatten[Table[If[a^b-b^a>-1&&a^b-b^a<10^6*2, a^b-b^a], {a, 1, 123}, {b, a, 144}]]] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2008 *)
nn=10^50; n=1; 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]]]
PROG
(PARI) list(lim)=my(v=List([0]), t); for(x=2, max(logint(lim\=1, 2)+1, 6), for(y=2, x-1, t=abs(x^y-y^x); if(t<=lim&&t, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jul 20 2017