OFFSET
1,2
COMMENTS
The next term, if it exists, is at least 2*10^17. - David Wasserman, May 01 2002
a(10) > 10^24, if it exists. The only numbers below 10^24 that can be written as m^k+m in more than one way are 30 = 5^2+5 = 3^3+3 and 130 = 5^3+5 = 2^7+2. - Giovanni Resta, Jun 21 2018
Conjectured to be finite and complete by Bennett (2001).
LINKS
Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Brady Haran and Matt Parker, Why 1980 was a great year to be born... but 2184 will be better, Numberphile video (2015).
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33. See Conjecture 3.
EXAMPLE
30 is in the sequence since 30 = 2^5 - 2 = 6^2 - 6;
2184 is in the sequence since 2184 = 3^7 - 3 = 13^3 - 13.
MAPLE
res:= {0}:
for k from 3 to 60 do
for m from 2 while m^k-m < 2^60 do
x:= m^k-m;
if assigned(R[x]) or issqr(4*x+1) then res:= res union {x}
else R[x]:= [m, k]
fi
od
od:
res; # Robert Israel, Oct 07 2015
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Henry Bottomley, Sep 26 2000
EXTENSIONS
More terms from Jud McCranie, Oct 01 2000
Offset corrected by Joerg Arndt, Oct 07 2015
STATUS
approved