Fifth power (algebra)

Short description: The result of multiplying five instances of a number together

In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together:

n⁵ = 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 n⁵ is the same as the last (decimal) digit of n, i.e.

[math]\displaystyle{ n \equiv n^5\pmod {10} }[/math]

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)

See also

• Eighth power
• Seventh power
• Sixth power
• Fourth power
• Cube (algebra)
• Square (algebra)
• Perfect power

References

• Råde, Lennart; Westergren, Bertil (2000) (in German). Springers mathematische Formeln: Taschenbuch für Ingenieure, Naturwissenschaftler, Informatiker, Wirtschaftswissenschaftler (3 ed.). Springer-Verlag. p. 44. ISBN 3-540-67505-1. https://books.google.com/books?id=DICwim5DphgC&q=44+Potenzen+n&pg=PA195. 
• Vega, Georg (1783) (in German). Logarithmische, trigonometrische, und andere zum Gebrauche der Mathematik eingerichtete Tafeln und Formeln. Vienna: Gedruckt bey Johann Thomas Edlen von Trattnern, kaiferl. königl. Hofbuchdruckern und Buchhändlern. p. 358. https://archive.org/details/logarithmischet00vongoog. "1 32 243 1024." 
• Jahn, Gustav Adolph (1839) (in German). 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. Leipzig: Verlag von Johann Ambrosius Barth. p. 241. https://books.google.com/books?id=BYA_AAAAcAAJ&q=1+32+243+1024&pg=PA241. 
• Deza, Elena; Deza, Michel (2012). Figurate Numbers. Singapore: World Scientific Publishing. p. 173. ISBN 978-981-4355-48-3. https://books.google.com/books?id=yJIMx9nXB6kC&q=%221%2C+32%2C+243%2C+1024%22&pg=PA159. 
• 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. https://books.google.com/books?id=yJIMx9nXB6kC&q=%221%2C+32%2C+243%2C+1024%22&pg=PA159. 
• Prändel, Johann Georg (1815) (in German). Arithmetik in weiterer Bedeutung, oder Zahlen- und Buchstabenrechnung in einem Lehrkurse - mit Tabellen über verschiedene Münzsorten, Gewichte und Ellenmaaße und einer kleinen Erdglobuslehre. Munich. p. 264. https://books.google.com/books?id=V_EoAAAAcAAJ&q=%221%2C+32%2C+243%2C+1024%22&pg=PA264.