TOPICS
Search

Euler's Conjecture

Euler's conjecture in Waring's problem asserts that, for every integer k>=2,

g(k)=2^k+|_(3/2)^k_|-2,

where g(k) is the least number such that every positive integer is a sum of at most g(k) kth powers of positive integers, and |_x_| is the floor function.

J. A. Euler, son of Leonhard Euler, established the corresponding lower bound (Waldschmidt 2015). Delmer and Deshouillers (1990, p. 886) suggest that he may also have expected equality, and use the name Euler's conjecture for this stronger assertion.

See also

Euler Quartic Conjecture, Euler's Graeco-Roman Squares Conjecture, Euler's Sum of Powers Conjecture, Waring's Problem

Explore with Wolfram|Alpha

References

Delmer, F. and Deshouillers, J.-M. "On the Computation of g(k) in Waring's Problem." Math. Comput. 54, 885-893, 1990. https://doi.org/10.1090/S0025-5718-1990-1011440-6.Waldschmidt, M. "Lecture on the abc Conjecture and Some of Its Consequences." In Mathematics in the 21st Century (Ed. P. Cartier, A. D. R. Choudary, and M. Waldschmidt). Basel, Switzerland: Springer, pp. 211-230, 2015. https://doi.org/10.1007/978-3-0348-0859-0_13.

Referenced on Wolfram|Alpha

Euler's Conjecture

Cite this as:

Weisstein, Eric W. "Euler's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulersConjecture.html

Subject classifications