Euler's sum of powers conjecture

Euler was aware of the equality involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. The general solution of the equation x_1^3+x_2^3=x_3^3+x_4^3 is \begin{align} x_1 &=\lambda( 1-(a-3b)(a^2+3b^2)) \\[2pt] x_2 &=\lambda( (a+3b)(a^2+3b^2)-1 )\\[2pt] x_3 &=\lambda( (a+3b)-(a^2+3b^2)^2 )\\[2pt] x_4 &= \lambda( (a^2+3b^2)^2-(a-3b)) \end{align} where , and {\lambda} are any rational numbers. == Counterexamples ==

Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for . This was published in a paper comprising just two sentences. His smallest counterexample was 20615673^4 = 2682440^4 + 15365639^4 + 18796760^4. A particular case of Elkies' solutions can be reduced to the identity (85v^2 + 484v - 313)^4 + (68v^2 - 586v + 10)^4 + (2u)^4 = (357v^2 - 204v + 363)^4, where u^2 = 22030 + 28849v - 56158v^2 + 36941v^3 - 31790v^4. This is an elliptic curve with a rational point at . From this initial rational point, one can compute an infinite collection of others. Substituting into the identity and removing common factors gives the numerical example cited above. In 1988, Roger Frye found the smallest possible counterexample 95800^4 + 217519^4 + 414560^4 = 422481^4 for by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000. == Generalizations ==

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if :\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k, where are positive integers for all and , then . In the special case , the conjecture states that if :\sum_{i=1}^{n} a_i^k = b^k (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 a^3 + b^3 = c^3. (The minimum positive value of a sum of third powers is 9^3 - 8^3 - 6^3 = 1 , 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 3^3 + 4^3 + 5^3 = 6^3 (Plato's number 216) This is the case , of Srinivasa Ramanujan's formula (3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3 A cube as the sum of three cubes can also be parameterized in one of two ways: (Lander, Parkin, Selfridge, smallest, 1967); ====== 568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 (M. Dodrill, 1999). ====== 1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8 (S. Chase, 2000). ==See also==