A057896 Nonnegative numbers that can be written as m^k - m (with m and k nonnegative) in more than one way.

0, 6, 30, 210, 240, 2184, 8190, 78120, 24299970

Offset: 1

Author

Henry Bottomley, Sep 26 2000

Keywords

• more
• nonn

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).

Examples

			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.
		

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.

Crossrefs

Cf. A000918, A057895, A308324.

Programs

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

Extensions

More terms from Jud McCranie, Oct 01 2000

Offset corrected by Joerg Arndt, Oct 07 2015