Hubbry Logo
91 (number)91 (number)Main
Open search
91 (number)
Community hub
91 (number)
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
91 (number)
91 (number)
91 (number)
View on Wikipedia
from Wikipedia
← 90 91 92 →
Cardinalninety-one
Ordinal91st
(ninety-first)
Factorization7 × 13
Divisors1, 7, 13, 91
Greek numeralϞΑ´
Roman numeralXCI, xci
Binary10110112
Ternary101013
Senary2316
Octal1338
Duodecimal7712
Hexadecimal5B16

91 (ninety-one) is the natural number following 90 and preceding 92.

Look up 9023112043 in Wiktionary, the free dictionary.

In mathematics

[edit]
91 is a square pyramidal number.
91 is a centered hexagonal number.
91 as the sum and difference of two positive cubes

91 is:

The decimal equivalent of the fraction 1⁄91 can be obtained by using powers of 9.

In science

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

91 (number)

View on Grokipedia
from Grokipedia
Ninety-one (91) is the natural number following 90 and preceding 92. It is an odd composite number with prime factorization 7×137 \times 13. In mathematics, 91 is the fourth centered cube number, given by the formula (2n−1)(n2−n+1)(2n-1)(n^2 - n + 1) for n=4n=4, representing the number of points in a cube with layers centered around a core point. It is also a Fermat pseudoprime to base 3, satisfying 390≡1(mod91)3^{90} \equiv 1 \pmod{91} despite being composite, and is pseudoprime to 35 bases less than itself. Additionally, the McCarthy 91-function, a recursive function defined by John McCarthy as M(n)=M(M(n+11))M(n) = M(M(n+11)) for n≤100n \leq 100 and M(n)=n−10M(n) = n - 10 for n>100n > 100, famously outputs 91 for all positive integer inputs from 1 to 101, illustrating key concepts in recursion and computability. Beyond , 91 holds significance in chemistry as the of (Pa), a rare, radioactive metal in the periodic table. In popular culture and curiosities, mathematician John H. Conway reportedly dubbed 91 the "smallest non-trivial prime" in jest, highlighting its composite nature while evading simple primality tests for small primes like 2, 3, and 5.

Mathematics

Arithmetic properties

91 is the product of two distinct prime numbers, 7 and 13, giving its prime as 91=7×1391 = 7 \times 13. The positive divisors of 91 are exactly 1, 7, 13, and 91. As a with exactly four divisors, 91 is not prime. Moreover, since its prime involves no repeated prime factors, 91 is square-free, meaning it is not divisible by any perfect square greater than 1. A notable arithmetic property of 91 arises in the context of primality testing via , which states that if pp is a and aa is an integer not divisible by pp, then ap−1≡1(modp)a^{p-1} \equiv 1 \pmod{p}. Despite being composite, 91 behaves like a prime under this test for base a=3a=3, satisfying 390≡1(mod91)3^{90} \equiv 1 \pmod{91}; such composites are called s to base 3. In fact, 91 is the smallest Fermat pseudoprime to base 3. The ϕ(n)\phi(n) counts the number of positive integers up to nn that are relatively prime to nn. For n=91=7×13n=91=7 \times 13, the formula ϕ(n)=n∏p∣n(1−1/p)\phi(n) = n \prod_{p \mid n} (1 - 1/p) yields ϕ(91)=91×(1−1/7)×(1−1/13)=91×(6/7)×(12/13)=72\phi(91) = 91 \times (1 - 1/7) \times (1 - 1/13) = 91 \times (6/7) \times (12/13) = 72.

Special representations and sequences

91 is the sixth square pyramidal number, which counts the number of spheres in a with a square base of six layers. This is given by the formula Pn=n(n+1)(2n+1)6P_n = \frac{n(n+1)(2n+1)}{6} evaluated at n=6n=6, yielding P6=6×7×136=91P_6 = \frac{6 \times 7 \times 13}{6} = 91. Equivalently, it is the sum of the squares of the first six natural numbers: 12+22+32+42+52+62=1+4+9+16+25+36=911^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91. Additionally, 91 is the sixth centered hexagonal number, representing the total dots in a with six rings around a central dot in a . The formula is Hn=3n(n−1)+1H_n = 3n(n-1) + 1, so for n=6n=6, H6=3×6×5+1=91H_6 = 3 \times 6 \times 5 + 1 = 91. An algebraic identity expresses 91 as both the sum and difference of cubes: 91=63+(−5)3=216−12591 = 6^3 + (-5)^3 = 216 - 125. This makes 91 the smallest positive integer writable as a sum of two nonzero cubes in two distinct ways, the other being 33+43=27+64=913^3 + 4^3 = 27 + 64 = 91. The reciprocal of 91 has a non-terminating expansion: 191=0.010989‾\frac{1}{91} = 0.\overline{010989}, with a repetend length (period) of 6. This period is the multiplicative order of 10 91, as 106≡1(mod91)10^6 \equiv 1 \pmod{91}. The repeating block "010989" derives from the process and relates to patterns in decimals of nearby fractions like 991=0.098901‾\frac{9}{91} = 0.\overline{098901} and 9991=1.087912‾\frac{99}{91} = 1.\overline{087912}, where digit sequences overlap with powers of 9 influencing the cyclic structure.

Science

Chemistry

(Pa) is the with 91 in the periodic table, belonging to the series of metals. It is a dense, silvery-gray metal that is highly radioactive and toxic due to its intense alpha-particle emission and chemical reactivity. The element was first identified in 1913 by Kasimir Fajans and Oswald Helmuth Göhring, who detected the short-lived ^{234m}Pa ( 1.17 minutes) as a in the series. In 1917, the longer-lived ^{231}Pa was independently discovered by and in , and by Frederick Soddy and John Arnold Cranston in the , leading to the naming of the element as proto-actinium (later shortened to ) to reflect its role as the parent of . The pure metallic form was first isolated in 1934 by Aristid von Grosse through of protactinium tetraiodide (PaI_4), following his earlier preparation of protactinium(V) oxide (Pa_2O_5) in 1927. Protactinium exhibits typical physical properties of metals, with an of 231.03588 u, a of 15.4 g/cm³, a of 1572°C, and a of 4000°C. Its is [Rn] 5f^2 6d^1 7s^2, reflecting the filling of the 5f orbitals characteristic of actinides. Chemically, is reactive, tarnishing rapidly in air to form an layer and reacting vigorously with , oxygen, and inorganic acids to produce gas and soluble salts. It displays oxidation states primarily of +4 and +5, forming compounds such as protactinium(IV) (PaO_2) and protactinium(IV) fluoride (PaF_4), which are stable in solid phases but hydrolyze in aqueous solutions. does not react with alkalis under normal conditions. Thirty radioisotopes of are known, ranging from ^{212}Pa to ^{239}Pa, all radioactive with no stable variants. The most stable and naturally occurring is ^{231}Pa, which has a of 32,760 years and decays via alpha emission to ^{227}Ac. This is significant in , serving as a key in thorium-uranium-protactinium (U-Th-Pa) methods for marine sediments, corals, and speleothems over timescales of 10^4 to 10^5 years. Protactinium occurs naturally in minute quantities as a of ^{235}U, primarily in ores such as pitchblende, at concentrations of approximately 1 part ^{231}Pa per 10 million parts of (up to 3 ppm in some ores from ). The ratio of to in these ores is approximately 1:3,000,000 by weight. Due to its extreme rarity, high radioactivity, and toxicity, has no significant commercial applications and is produced only in quantities for research. Its primary scientific value lies in studies of chemistry, particularly the behavior of 5f electrons, and in nuclear research related to the series and transuranic elements.

Astronomy

Messier 91 (M91), also designated NGC 4548, is a of morphological type SBb situated in the constellation as a member of the . Approximately 60 million light-years distant from Earth, it exhibits an of 11.0 and an angular size of 5.4' × 4.3'. The galaxy was observed on April 29, 1781, by during his survey of the , but its recorded coordinates were inaccurate—likely due to an error in referencing nearby Messier objects M58 and M89—rendering it a "lost" entry in the catalog for nearly two centuries. In 1969, amateur astronomer William C. Williams resolved the discrepancy by matching it to NGC 4548, which had been cataloged by in 1784. M91 displays a conspicuous central bar extending into tightly wound spiral arms populated by clusters of young, blue stars and interrupted by prominent dark dust lanes, culminating in a bright nucleus powered by a . Though classified as an anemic with relatively low gas content and subdued star formation compared to typical spirals, observations capture its detailed bar structure and arm features, underscoring its role in studies of dynamics. As one of the faintest Messier objects, M91 poses a challenge for amateur telescopes, often requiring apertures of 8 inches or larger under to discern its faint glow and elongated form. NGC 91 is a faint, small in the constellation Andromeda, with an apparent magnitude of 14.5, a stellar nucleus, and broad, weak arms; it belongs to a that includes NGC 80, NGC 83, NGC 90, and NGC 93.

Other uses

Transportation

( 91) is a north-south in the , currently spanning 172.7 miles from , to . Established in 1926 as part of the original U.S. Highway system, it originally extended over 1,400 miles from , to the Canadian border at Sweetgrass, , serving as a vital transcontinental corridor. The route largely follows the historic Arrowhead Trail, the first all-weather road connecting to , and incorporates segments of the used by settlers in the mid-19th century. Paved in the , 91 became the first fully paved highway through southern , facilitating early automobile travel and commerce along its path. Over time, much of the original alignment was superseded by in the 1970s, truncating US 91 to its current length and earning it the nickname "Old Highway 91" in , where it remains a scenic byway dotted with historic markers and tourist attractions. The highway plays a key role in the , supporting for and industry while drawing tourists to natural sites like Bear Lake and the . Interstate 91 (I-91) is a 290-mile north-south interstate in , running from , to the Canadian border at , and paralleling the for much of its length. Constructed primarily between 1958 and the early 1980s as part of the , it connects urban centers like Hartford and Springfield to rural areas, forming a crucial link in the Northeast's transportation network. In urban sections, such as near Hartford, I-91 carries approximately 152,000 vehicles daily (as of 2015), underscoring its importance for , , and regional travel. Several U.S. states designate Route 91 as a major local highway; for example, is a busy east-west freeway spanning about 59 miles through the area, from Beach Boulevard in Anaheim to Interstate 215 in Riverside, serving as a vital artery for commuters between Riverside County and Orange County with express lanes to manage congestion.

Sports

In , the number 91 holds particular significance in team sports like hockey, , and , where it has been worn by standout athletes known for their defensive prowess and championship contributions. In the National Hockey League (NHL), Sergei Fedorov epitomizes the number's legacy as the longtime center for the Detroit Red Wings, wearing #91 from 1990 to 2004 during his tenure with the team. Fedorov led the Red Wings to three Stanley Cup victories in 1997, 1998, and 2002, and he earned the Hart Memorial Trophy as the league's most valuable player in 1994. On August 19, 2025, the Red Wings announced the retirement of Fedorov's #91 jersey, with the on-ice ceremony set for January 12, 2026, against the Carolina Hurricanes. His enduring impact has inspired other NHL stars to adopt the number, including Toronto Maple Leafs captain John Tavares, who has worn #91 throughout his professional career and credits Fedorov's influence for his choice. Similarly, Steven Stamkos, a multiple-time All-Star and former captain of the Tampa Bay Lightning, has donned #91 since entering the league in 2008, amassing over 900 points and two Stanley Cups in 2020 and 2021. In the (NBA), wore #91 as a forward for the from 1995 to 1998, a period in which he helped secure three NBA championships as part of the team's dynasty. Earlier with the , Rodman won two additional titles in 1989 and 1990, bringing his total to five championships, and he captured the NBA Defensive Player of the Year award three consecutive times from 1990 to 1992 for his rebounding dominance. Rodman selected #91 to symbolize his distinctive and rebellious persona, setting it apart from conventional jersey numbers. In the (), the number 91 has been associated with defensive linemen for the , notably , who wore it as a from 2009 to 2018 and recorded 100.5 career sacks, ranking second in franchise history. also donned #91 for the Dolphins from 2005 to 2008, contributing 14.5 sacks during his stint, including a career-high 7.0 in 2006. Fedorov's prominence has cemented #91's status in hockey culture, where it often evokes themes of versatility and excellence.

Culture and history

In religious contexts, from the Book of Psalms holds particular significance as the "Psalm of Protection," offering assurances of divine safeguarding for the faithful. The psalm opens with the verse, "He who dwells in the shelter of the Most High will rest in the shadow of the Almighty," and proceeds to promise deliverance from perils such as plagues, arrows, and nocturnal terrors, emphasizing God's role as a refuge and fortress. This text is recited in Jewish liturgies, including during and personal prayers for safety, and in Christian traditions, where it provides comfort during crises and is incorporated into services for protection. In and symbolism, the number 91 is interpreted as reducing to 1 through digit (9 + 1 = 10, then 1 + 0 = 1), representing , , and the onset of new beginnings. As an angel number, 91 is seen as a encouraging transitions, humanitarian efforts, and trust in spiritual guidance to manifest positive change. Biblically, its as 7 × 13 symbolizes the interplay of (7) and rebellion or (13), adding layers of interpretive depth in theological . Historically, the year 91 BC marked the outbreak of the Social War in the Roman Republic, a rebellion by Italian allies (socii) denied full citizenship, who sought independence and equal rights after the assassination of reformer Marcus Livius Drusus. Key Roman figures, including generals Gaius Marius and Lucius Cornelius Sulla, led the military response against the rebels, with the conflict ultimately resulting in the enfranchisement of most Italians by 89 BC through laws like the Lex Julia and Lex Plautia Papiria. In contrast, AD 91 was a relatively uneventful common year in the Julian calendar under Emperor Domitian, notable primarily for the consulship shared by Manius Acilius Glabrio and Marcus Ulpius Trajan, the latter who would later ascend as emperor in 98 AD, amid routine administrative affairs without recorded major battles. Appearances of 91 in broader culture, such as and , are infrequent, though it is linked to its pseudoprime properties, where it mimics prime behavior in specific tests despite being composite (7 × 13).

References

  1. https://collegedunia.com/exams/factors-of-91-mathematics-articleid-7377
  2. https://mathworld.wolfram.com/CenteredCubeNumber.html
  3. https://math.stackexchange.com/questions/67924/how-to-show-that-91-is-a-pseudoprime-to-the-base-3
  4. https://t5k.org/curios/page.php/91.html
  5. https://mathworld.wolfram.com/McCarthy91-Function.html
  6. https://pubchem.ncbi.nlm.nih.gov/element/Protactinium
  7. http://www.mrob.com/pub/math/numbers-9.html
  8. https://www.cuemath.com/numbers/factors-of-91/
  9. https://byjus.com/maths/is-91-a-prime-number/
  10. https://mathworld.wolfram.com/Squarefree.html
  11. https://mathworld.wolfram.com/FermatPseudoprime.html
  12. https://www.scientificamerican.com/blog/roots-of-unity/91-is-april-fooling-you/
  13. https://mathworld.wolfram.com/TotientFunction.html
  14. https://mathworld.wolfram.com/SquarePyramidalNumber.html
  15. https://mathworld.wolfram.com/HexNumber.html
  16. https://oeis.org/A003215
  17. https://oeis.org/A001235
  18. https://oeis.org/A225908
  19. https://oeis.org/A051626
  20. https://calculat.io/en/number/fraction-as-a-decimal/0--1--91
  21. https://periodic-table.rsc.org/element/91/protactinium
  22. https://periodic.lanl.gov/91.shtml
  23. https://www.osti.gov/servlets/purl/782642
  24. https://www.lenntech.com/periodic/elements/pa.htm
  25. https://www.webelements.com/protactinium/compounds.html
  26. https://www.ebsco.com/research-starters/science/protactinium-pa
  27. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JB073i006p02271
  28. http://hpschapters.org/northcarolina/NSDS/Protactinium.pdf
  29. https://science.nasa.gov/mission/hubble/science/explore-the-night-sky/hubble-messier-catalog/messier-91/
  30. https://www.messier-objects.com/messier-91/
  31. https://noirlab.edu/public/images/noao-m91/
  32. http://www.messier.seds.org/m/m091.html
  33. https://www.sci.news/astronomy/hubble-image-messier-91-10695.html
  34. http://www.docdb.net/show_object.php?id=ngc_91
  35. https://gbcnet.com/ushighways/US91.html
  36. https://www.usends.com/91.html
  37. https://history.utah.gov/arrowhead-trail-aka-old-us-91-highway/
  38. https://www.blm.gov/visit/old-highway-91
  39. https://i15alliance.org/Documents/I-15_CSMP_v31.pdf
  40. https://vermonthistory.org/freedom-unity-coming-of-the-interstate
  41. https://crcogct.gov/wp-content/uploads/2016/12/Hartford_TMA_CMP_2015_Final.pdf
  42. https://dot.ca.gov/-/media/dot-media/district-12/documents/f0007313-route91.pdf
  43. https://www.thetorah.com/article/god-shelters-the-faithful-the-prayer-of-psalm-91
  44. https://biblehub.com/topical/p/psalm_91.htm
  45. https://affinitynumerology.com/number-meanings/number-91-meaning.php
  46. https://www.sunsigns.org/angel-number-91-meaning/
  47. https://www.biblestudy.org/bibleref/meaning-of-numbers-in-bible/91.html
  48. https://www.britannica.com/event/Social-War-Roman-history
  49. https://www.historyhit.com/timeline-marius-and-sulla/
  50. https://www.britannica.com/biography/Trajan
Add your contribution
Related Hubs
User Avatar
No comments yet.