A045575 Nonnegative numbers of the form x^y - y^x, for x,y > 1.

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

Offset: 1

Author

Erich Friedman

Keywords

• easy
• nonn

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.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 2009.

Crossrefs

Cf. A076980.

Programs

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:
{A}; # Robert Israel, Aug 20 2014

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
				

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