Fourth power

Properties

The last digit of a fourth power in decimal can only be 0, 1, 5, or 6.

In hexadecimal the last nonzero digit of a fourth power is always 1.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem).

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:

: .

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:Quoted in | access-date = 17 July 2017

: (Allan MacLeod) : (D.J. Bernstein) : (D.J. Bernstein) : (D.J. Bernstein) : (D.J. Bernstein) : (Roger Frye, 1988) : (Allan MacLeod, 1998)

Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using radicals.

References

References

1. An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)² = 64n² + 16n + 1 = 16(4n² + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2^kn)⁴ = 16^kn⁴ for some positive integer k and odd integer n, meaning that an even fourth power can be represented as an odd fourth power multiplied by a power of 16.