Number

33

33 — The master number

33 is a composite number, odd, a calendar year.

33 carries an unusual freight of significance for so small a number. It is the traditional age of Jesus at the crucifixion; the highest degree of the Scottish Rite of Freemasonry; the number of vertebrae an infant is born with before some fuse; and the count of cantos Dante gives each of Purgatorio and Paradiso. Numerologists call it a "master number," and it has attached itself to everything from the label of a beer to conspiracy folklore.

Its most striking modern chapter, though, is pure mathematics. For decades 33 sat on a short list of stubborn numbers: which integers can be written as a sum of three cubes, a³ + b³ + c³? For 33 nobody knew, and it became — with 42 — one of the last two values under 100 left unsolved. In 2019 the mathematician Andrew Booker finally cracked it, with a global computer search turning up

\(33 = 8\,866\,128\,975\,287\,528^3 - 8\,778\,405\,442\,862\,239^3 - 2\,736\,111\,468\,807\,040^3\)

three sixteen-digit numbers that improbably collapse to 33. Its solution left just one holdout below 100 — the famous 42, cleared a few months later. A number heavy with mystic meaning turned out to hide a genuinely hard secret in plain arithmetic.

Sources https://en.wikipedia.org/wiki/Sums_of_three_cubes · https://en.wikipedia.org/wiki/33_(number)

Arithmetic Number Binary Palindrome Cube-Free Curated Deficient Number Evil Number Palindrome Pernicious Number Preferred Number Recamán's Sequence Repdigit Semiprime Squarefree

Interestingness

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9 notable facts · grade A

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Historical context — 33 AD

Calendar year

AD 33 (XXXIII) was a common year starting on Thursday of the Julian calendar.

Excerpt from Wikipedia (en) ↗ · Licensed CC BY-SA 4.0 Read the full article on Wikipedia →

Historical context — 33 BC

Calendar year

Year 33 BC was either a common year starting on Saturday, Sunday or Monday or a leap year starting on Sunday of the Julian calendar and a leap year starting on Saturday of the Proleptic Julian calendar.

Excerpt from Wikipedia (en) ↗ · Licensed CC BY-SA 4.0 Read the full article on Wikipedia →

Year facts

Year type: Common year
Standard 365-day year; not divisible by 4 (or divisible by 100 but not 400).
Days in year: 365
ISO weeks: 52
Started on: Saturday
January 1, 33
Ended on: Saturday
December 31, 33
Friday the 13ths: 1
One Friday the 13th this year.
Decade: 30s
30–39
Century: 1st century
1–100
Millennium: 1st millennium
1–1000
Years ago: 1,993
1993 years before 2026.

In other calendars

Hebrew: 3793 / 3794 AM
Rosh Hashanah falls in September/October.
Chinese: Year of the zodiac:Water zodiac:Snake
Sexagenary cycle position 30 of 60. Lunar new year falls in late January / mid-February.
Buddhist Era: 576 BE
Counted from the parinirvana of the Buddha (Theravada / Thai / Sri Lankan convention).
Ethiopian: 25 / 26 ET
Year boundary at Enkutatash (September 11/12).
Indian National (Saka): -45 / -46 Saka
Indian national calendar; year starts in March.

Cultural significance

Christian significant

Traditional age of Jesus at his crucifixion.

Freemasonry significant

Highest degree in the Scottish Rite of Freemasonry.

Sourced from Wikipedia (Numerology, Chinese numerology, Gematria, and per-culture articles).

Properties

Parity: Odd
Digit count: 2
Digit sum: 6
Digit product: 9
Digital root: 6
Palindrome: Yes
Bit width: 6 bits
Recamán's sequence: a(50) = 33
Square (n²): 1,089
Cube (n³): 35,937
Divisor count: 4
σ(n) — sum of divisors: 48
φ(n) — Euler's totient: 20
Sum of prime factors: 14

Primality

Prime factorization: 3 × 11

Nearest primes: 31 (−2) · 37 (+4)

Divisors & multiples

All divisors (4)

1 · 3 · 11 · 33

Aliquot sum (sum of proper divisors): 15

Factor pairs (a × b = 33)

1 × 33

3 × 11

First multiples

33 · 66 (double) · 99 · 132 · 165 · 198 · 231 · 264 · 297 · 330

Sums & aliquot sequence

As consecutive integers: 16 + 17 10 + 11 + 12 3 + 4 + 5 + 6 + 7 + 8

Aliquot sequence: 33 → 15 → 9 → 4 → 3 → 1 → 0 — terminates at zero

Continued fraction of √n

√33 = [5; (1, 2, 1, 10)]

Period length 4 — the block in parentheses repeats forever.

Representations

In words: thirty-three
Ordinal: 33rd
Roman numeral: XXXIII
Binary: 100001
Octal: 41
Hexadecimal: 0x21
Base64: IQ==
One's complement: 222 (8-bit)
Scientific notation: 3.3 × 10¹

In other bases

ternary (3) 1020

quaternary (4) 201

quinary (5) 113

senary (6) 53

septenary (7) 45

nonary (9) 36

undecimal (11) 30

duodecimal (12) 29

tridecimal (13) 27

tetradecimal (14) 25

pentadecimal (15) 23

Palindromic in base 2

As an angle

33°

33° ≈ 0.576 rad

Compass bearing: NNE (north-northeast)

Why does a circle have 360 degrees? →

Historical numeral systems

Babylonian (base 60): 𒌋𒌋𒌋𒁹𒁹𒁹
Egyptian hieroglyphic: 𓎆𓎆𓎆𓏺𓏺𓏺
Greek (Milesian): λγʹ
Mayan (base 20): 𝋡·𝋭
Chinese: 三十三
Chinese (financial): 參拾參

In other modern scripts

Eastern Arabic ٣٣ Devanagari ३३ Bengali ৩৩ Tamil ௩௩ Thai ๓๓ Tibetan ༣༣ Khmer ៣៣ Lao ໓໓ Burmese ၃၃

Digit at this position in famous constants

π — Pi (π): Digit 33 = 0
e — Euler's number (e): Digit 33 = 6
φ — Golden ratio (φ): Digit 33 = 3
√2 — Pythagoras's (√2): Digit 33 = 9
ln 2 — Natural log of 2: Digit 33 = 7
γ — Euler-Mascheroni (γ): Digit 33 = 0

Also seen as

ASCII character

As an ASCII codepoint, 33 is !. Printable ASCII character !.

Hex color

#000021

RGB(0, 0, 33)

IPv4 address

As an unsigned 32-bit integer, this is the IPv4 address 0.0.0.33.

Address: 0.0.0.33
Class: reserved
IPv4-mapped IPv6: ::ffff:0.0.0.33

Unspecified address (0.0.0.0/8) — "this network" placeholder.

Musical pitch

Heard as a frequency, 33 Hz is closest to:

Concert pitch (A4 = 440 Hz): C1 (32.7 Hz, +16¢)
Scientific pitch (C4 = 256 Hz): C♯1 (33.9 Hz, -47¢ — about midway to C1)
Baroque pitch (A4 = 415 Hz): C♯1 (32.7 Hz, +17¢)

(very low — many laptop and phone speakers can't reproduce it)

As a MIDI note number, 33 is A1 (55 Hz at concert pitch). (very low — many laptop and phone speakers can't reproduce it)

Atomic number

On the periodic table, atomic number 33 is Arsenic (As) — period 4.

Arsenic on Wikipedia ↗