25 (number)

← 24 25 26 →
← 20 21 22 23 24 25 26 27 28 29 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-five
Ordinal	25th (twenty-fifth)
Factorization	52
Divisors	1, 5, 25
Greek numeral	ΚΕ´
Roman numeral	XXV, xxv
Binary	110012
Ternary	2213
Senary	416
Octal	318
Duodecimal	2112
Hexadecimal	1916

25 (twenty-five) is the natural number following 24 and preceding 26.

In mathematics

25 is a square.

It is a square number, being 5 ² = 5 × 5, and hence the third non-unitary square prime of the form p².

It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 25² = 625; the other is 76.

25 has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in (1 and 0).

It is the smallest square that is also a sum of two (non-zero) squares: 25 = 3² + 4². Hence, it often appears in illustrations of the Pythagorean theorem.

25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9.

25 is a centered octagonal number, ^[1] a centered square number, ^[2] a centered octahedral number, ^[3] and an automorphic number. ^[4]

25 percent (%) is equal to ⁠1/4⁠.

It is the smallest decimal Friedman number as it can be expressed by its own digits: 5². ^[5]

It is also a Cullen number ^[6] and a vertically symmetrical number. ^[7] 25 is the smallest pseudoprime satisfying the congruence 7ⁿ = 7 mod n.

25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. ^[8]

According to the Shapiro inequality, 25 is the smallest odd integer n such that there exist x₁, x₂, ..., x_n such that

${\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}<{\frac {n}{2}}}$

where x_{n + 1} = x₁, x_{n + 2} = x₂.

Within decimal, one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00, 25, 50, or 75.

There are 25 primes under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

F₄, H₄ symmetry and lattices Λ₂₄, II_25,1

Twenty-five 24-cells with ${\displaystyle \mathrm {F_{4}} }$ symmetry in the fourth dimension can be arranged in two distinct manners, such that

• in a 24-cell honeycomb, twenty-four 24-cells surround a single 24-cell, and where
• a faceting of the 600-cell with ${\displaystyle \mathrm {H_{4}} }$ symmetry can otherwise also be constructed, with cells overlapping. ^[9]

The 24-cell can be further generated using three copies of the 8-cell, where the 24-cell honeycomb is dual to the 16-cell honeycomb (with the tesseract the dual polytope to the 16-cell).

On the other hand, the positive unimodular lattice ${\displaystyle \mathrm {II_{25,1}} }$ in twenty-six dimensions is constructed from the Leech lattice in twenty-four dimensions using Weyl vector ^[10]

${\displaystyle (0,1,2,3,4,\ldots ,24|70)}$

that features the only non-trivial solution, i.e. aside from ${\displaystyle \{0,1\}}$, to the cannonball problem where sum of the squares of the first twenty-five natural numbers ${\displaystyle \{0,1,2,\ldots ,24\}}$ in ${\displaystyle \mathbb {N_{0}} }$ is in equivalence with the square of ${\displaystyle 70}$ ^[11] (that is the fiftieth composite). ^[12] The Leech lattice, meanwhile, is constructed in multiple ways, one of which is through copies of the ${\displaystyle \mathbb {E_{8}} }$ lattice in eight dimensions ^[13] isomorphic to the 600-cell, ^[14] where twenty-five 24-cells fit; a set of these twenty-five integers can also generate the twenty-fourth triangular number, whose value twice over is ${\displaystyle 600=24\times 25.}$ ^[15]

In religion

• In Ezekiel's vision of a new temple: The number twenty-five is of cardinal importance in Ezekiel's Temple Vision (in the Bible, Ezekiel chapters 40–48). ^[16]

In sports

• In baseball, the number 25 is typically reserved for the best slugger on the team. Examples include Mark McGwire, Barry Bonds, Jim Thome, and Mark Teixeira.

In other fields

Twenty-five is:

• The number of years of marriage marked in a silver wedding anniversary.

References

1. Sloane, N. J. A. (ed.). "SequenceA016754(Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA001844(Centered square numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Sloane, N. J. A. (ed.). "SequenceA001845(Centered octahedral numbers (crystal ball sequence for cubic lattice))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. Sloane, N. J. A. (ed.). "SequenceA003226(Automorphic numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA036057(Friedman numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. Sloane, N. J. A. (ed.). "SequenceA002064(Cullen numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
7. Sloane, N. J. A. (ed.). "SequenceA053701(Vertically symmetric numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA063769(Aspiring numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Denney, Tomme; Hooker, Da’Shay; Johnson, De’Janeke; Robinson, Tianna; Butler, Majid; Sandernisha, Claiborne (2020). "The geometry of H4 polytopes". Advances in Geometry . 20 (3). Berlin: De Gruyter: 433–444. arXiv: 1912.06156 . doi:10.1515/advgeom-2020-0005. S2CID   220367622.
10. Sloane, N. J. A. (ed.). "SequenceA351831(Vector in the 26-dimensional even Lorentzian unimodular lattice II_25,1 used to construct the Leech lattice.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-03-12.
11. Conway, John H. (1999). "Chapter 26: Lorentzian forms for the Leech lattice" . Sphere packings, lattices, and groups. Grundlehren der mathematischen Wissenschaften. Vol. 290 (1st ed.). New York: Springer. pp. 524–528. doi:10.1007/978-1-4757-6568-7. ISBN   978-0-387-98585-5. MR   1662447. OCLC   854794089.
12. Sloane, N. J. A. (ed.). "SequenceA002808(The composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-03-12.
13. Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN   2196-9701. ISBN   978-1-4757-2016-7. MR   1541550.
14. Baez, John C. (2018). "From the Icosahedron to E₈". London Mathematical Society Newsletter. 476: 18–23. arXiv: 1712.06436 . MR   3792329. S2CID   119151549. Zbl   1476.51020.
15. Sloane, N. J. A. (ed.). "SequenceA002378(Oblong (or promic, pronic, or heteromecic) numbers: a(n) equal to n*(n+1).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-03-16.
16. "Number 25 meaning in the Bible". Bible Wings. 2023-07-21. Retrieved 2023-11-02.