Fifth power (algebra)

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:

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.

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]

Footnotes

^ "Webster's 1913".
^ Lander, L. J.; Parkin, T. R. (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72 (6): 1079. doi:10.1090/S0002-9904-1966-11654-3.

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.

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
- Multiplicative

Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy