# 25 (number)

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

 ← 24 25 26 →
Cardinaltwenty-five
Ordinal25th
(twenty-fifth)
Factorization52
Divisors1, 5, 25
Greek numeralΚΕ´
Roman numeralXXV
Binary110012
Ternary2213
Senary416
Octal318
Duodecimal2112

## In mathematics

It is a square number, being 52 = 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.

Twenty five 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 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, ..., xn such that

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

where xn + 1 = x1, xn + 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, II25,1

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).[18]
• In Islam, there are 25 prophets mentioned in the Quran.

## In sports

• Before 2020, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expanded their roster to 40 players.
• The size of the playing roster on a Nippon Professional Baseball team for a particular game. Active NPB rosters consist of 28 players, but prior to each game, managers must designate three players who will be ineligible for that game.
• 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.
• The number of points needed to win a set in volleyball under rally scoring rules (except for the fifth set), so long as the losing team's score is two less than the winning team's score (i.e., if the winning team scores 25 points, the losing team can have no more than 23 points)
• In U.S. college football, schools that are members of NCAA Division I FBS are allowed to provide athletic scholarships to a maximum of 25 new football players (i.e., players who were not previously receiving scholarships) each season.

Twenty-five is:

## Slang names

• Pony (British slang for £25)[19]

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A016754 (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.). "Sequence A001844 (Centered square numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^
4. ^ Sloane, N. J. A. (ed.). "Sequence A003226 (Automorphic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. ^ Sloane, N. J. A. (ed.). "Sequence A036057 (Friedman numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. ^ Sloane, N. J. A. (ed.). "Sequence A002064 (Cullen numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. ^ Sloane, N. J. A. (ed.). "Sequence A053701 (Vertically symmetric numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. ^ Sloane, N. J. A. (ed.). "Sequence A063769 (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.). "Sequence A351831 (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.). "Sequence A002808 (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.). "Sequence A002378 (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. ^ Meija, Juris; Coplen, Tyler B.; et al. (March 1, 2016). "Atomic weights of the elements 2013 (IUPAC Technical Report)". Pure and Applied Chemistry. 88 (3): 265–291. doi:10.1515/pac-2015-0305. hdl:11858/00-001M-0000-0029-C3D7-E. ISSN 0033-4545. S2CID 101719914.
17. ^ "Understanding Genetics". genetics.thetech.org. Retrieved 2 April 2018.
18. ^ "Number 25 meaning in the Bible". Bible Wings. 2023-07-21. Retrieved 2023-11-02.
19. ^ Evans, I.H., Brewer's Dictionary of Phrase and Fable, 14th ed., Cassell, 1990, ISBN 0-304-34004-9