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Fourth power
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==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.<ref>An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)<sup>2</sup> = 64n<sup>2</sup> + 16n + 1 = 16(4n<sup>2</sup> + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2<sup>k</sup>n)<sup>4</sup> = 16<sup>k</sup>n<sup>4</sup> 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.</ref> 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]]). [[Pierre de Fermat|Fermat]] knew that a fourth power cannot be the sum of two other fourth powers (the [[Proof of Fermat's Last Theorem for specific exponents#n = 4|''n'' = 4 case]] of [[Fermat's Last Theorem]]; see [[Fermat's right triangle theorem]]). [[Euler]] [[Euler's sum of powers conjecture|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 [[Noam Elkies|Elkies]] with: : {{math|1=20615673<sup>4</sup> = 18796760<sup>4</sup> + 15365639<sup>4</sup> + 2682440<sup>4</sup>}}. Elkies showed that there are infinitely many other [[counterexample]]s for exponent four, some of which are:<ref name=meyrignac>Quoted in {{cite web | last = Meyrignac | first = Jean-Charles | url = http://euler.free.fr/records.htm | title = Computing Minimal Equal Sums Of Like Powers: Best Known Solutions | date = 14 February 2001 | access-date = 17 July 2017 }}</ref> :{{math|1=2813001<sup>4</sup> = 2767624<sup>4</sup> + 1390400<sup>4</sup> + 673865<sup>4</sup>}} (Allan MacLeod) :{{math|1= 8707481<sup>4</sup> = 8332208<sup>4</sup> + 5507880<sup>4</sup> + 1705575<sup>4</sup>}} (D.J. Bernstein) :{{math|1=12197457<sup>4</sup> = 11289040<sup>4</sup> + 8282543<sup>4</sup> + 5870000<sup>4</sup>}} (D.J. Bernstein) :{{math|1=16003017<sup>4</sup> = 14173720<sup>4</sup> + 12552200<sup>4</sup> + 4479031<sup>4</sup>}} (D.J. Bernstein) :{{math|1=16430513<sup>4</sup> = 16281009<sup>4</sup> + 7028600<sup>4</sup> + 3642840<sup>4</sup>}} (D.J. Bernstein) :{{math|1= 422481<sup>4</sup> = 414560<sup>4</sup> + 217519<sup>4</sup> + 95800<sup>4</sup>}} (Roger Frye, 1988) :{{math|1= 638523249<sup>4</sup> = 630662624<sup>4</sup> + 275156240<sup>4</sup> + 219076465<sup>4</sup>}} (Allan MacLeod, 1998)
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