conjectured that if
:,where are positive integers for all and , then . In the special case , the conjecture states that if
:(under the conditions given above) then .

The special case may be described as the problem of giving a partition of a perfect power into few like powers. For and or , there are many known solutions. Some of these are listed below.

See for more data.

From Fermat's Last Theorem, we know that there can't be a solution to .(The minimum positive value of a sum of third powers is , which provides a solution to the equation (a = (1, 6, 8), b = 9), where however the smallest member isn't larger than 1.)The smallest solution with terms > 1 is (Plato's number 216)This is the case , of Srinivasa Ramanujan's formula A cube as the sum of three cubes can also be parameterized in one of two ways:The number 2,100,000³ can be expressed as the sum of three positive cubes in nine different ways.

(R. Frye, 1988); (R. Norrie, smallest, 1911).

(Lander & Parkin, 1966); (Lander, Parkin, Selfridge, smallest, 1967); (Lander, Parkin, Selfridge, second smallest, 1967); (Sastry, 1934, third smallest).

It has been known since 2002 that there are no solutions for whose final term is <= 730000.

(M. Dodrill, 1999).

(S. Chase, 2000).