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Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986, this was disproven by Elkies with: 20615673 4 = 18796760 4 ...
The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power. The word "raised" is usually omitted, and sometimes "power" as well, so 3 5 can be simply read "3 to the 5th", or "3 to the 5".
James Waldby, A Table of Fifth Powers equal to a Fifth Power (2009) R. Gerbicz, J.-C. Meyrignac, U. Beckert, All solutions of the Diophantine equation a 6 + b 6 = c 6 + d 6 + e 6 + f 6 + g 6 for a,b,c,d,e,f,g < 250000 found with a distributed Boinc project; EulerNet: Computing Minimal Equal Sums Of Like Powers; Weisstein, Eric W. "Euler's Sum ...
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation a n + b n = c n for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b. For n equal to 2, the equation has infinitely many solutions, the Pythagorean triples.)
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
The term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. [3] It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986. The term hyperpower [4] is a natural combination of hyper and power, which aptly describes tetration.
The sum of the reciprocals of the cubes of positive integers is called Apéry's constant ζ(3) , and equals approximately 1.2021 . This number is irrational, but it is not known whether or not it is transcendental. The reciprocals of the non-negative integer powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any ...
Semi-log plot of solutions of + + = for integer , , and , and .Green bands denote values of proven not to have a solution.. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.