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The result of multiplying five instances of a number together
In
arithmetic and
algebra , the fifth
power or sursolid
[1] of a
number n is the result of multiplying five instances of n together:
n 5 = n × n × n × n × n .
Fifth powers are also formed by multiplying a number by its
fourth power , or the
square of a number by its
cube .
The sequence of fifth powers of
integers is:
0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, ... (sequence
A000584 in the
OEIS )
Properties
For any integer n , the last
decimal digit of n 5 is the same as the last (decimal) digit of n , i.e.
n
≡
n
5
(
mod
10
)
{\displaystyle n\equiv n^{5}{\pmod {10}}}
By the
Abel–Ruffini theorem , there is no general
algebraic formula (formula expressed in terms of
radical expressions ) for the solution of
polynomial equations containing a fifth power of the
unknown as their highest power. This is the lowest power for which this is true. See
quintic equation ,
sextic equation , and
septic equation .
Along with the fourth power, the fifth power is one of two powers k that can be expressed as the sum of k − 1 other k -th powers, providing counterexamples to
Euler's sum of powers conjecture . Specifically,
275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966)
[2]
See also
References
Råde, Lennart; Westergren, Bertil (2000).
Springers mathematische Formeln: Taschenbuch für Ingenieure, Naturwissenschaftler, Informatiker, Wirtschaftswissenschaftler (in German) (3 ed.). Springer-Verlag. p. 44.
ISBN
3-540-67505-1 .
Vega, Georg (1783).
Logarithmische, trigonometrische, und andere zum Gebrauche der Mathematik eingerichtete Tafeln und Formeln (in German). Vienna: Gedruckt bey Johann Thomas Edlen von Trattnern, kaiferl. königl. Hofbuchdruckern und Buchhändlern. p.
358 . 1 32 243 1024.
Jahn, Gustav Adolph (1839).
Tafeln der Quadrat- und Kubikwurzeln aller Zahlen von 1 bis 25500, der Quadratzahlen aller Zahlen von 1 bis 27000 und der Kubikzahlen aller Zahlen von 1 bis 24000 (in German). Leipzig: Verlag von Johann Ambrosius Barth. p. 241.
Deza, Elena ; Deza, Michel (2012).
Figurate Numbers . Singapore: World Scientific Publishing. p. 173.
ISBN
978-981-4355-48-3 .
Rosen, Kenneth H.; Michaels, John G. (2000).
Handbook of Discrete and Combinatorial Mathematics . Boca Raton, Florida: CRC Press. p. 159.
ISBN
0-8493-0149-1 .
Prändel, Johann Georg (1815).
Arithmetik in weiterer Bedeutung, oder Zahlen- und Buchstabenrechnung in einem Lehrkurse - mit Tabellen über verschiedene Münzsorten, Gewichte und Ellenmaaße und einer kleinen Erdglobuslehre (in German). Munich. p. 264.
Possessing a specific set of other numbers
Expressible via specific sums