61 (number)

For the Cyrillic letter, see Ы.

← 60 61 62 →
← 60 61 62 63 64 65 66 67 68 69 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	sixty-one
Ordinal	61st (sixty-first)
Factorization	prime
Prime	18th
Divisors	1, 61
Greek numeral	ΞΑ´
Roman numeral	LXI
Binary	1111012
Ternary	20213
Senary	1416
Octal	758
Duodecimal	5112
Hexadecimal	3D16

61 (sixty-one) is the natural number following 60 and preceding 62.

In mathematics

61 is the 18th prime number, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, ${\displaystyle 5^{2}+6^{2}}$. ^[1] It is also a centered decagonal number, ^[2] and a centered hexagonal number. ^[3]

61 is the fourth cuban prime of the form ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$ where ${\displaystyle x=y+1}$, ^[4] and the fourth Pillai prime since ${\displaystyle 8!+1}$ is divisible by 61, but 61 is not one more than a multiple of 8. ^[5] It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ... ^[6]

61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (141₆) and 60 (11₆₀). It is the sixth up/down or Euler zigzag number.

61 is the smallest proper prime, a prime ${\displaystyle p}$ which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length ${\displaystyle p-1,}$ where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, ${\displaystyle {\tfrac {p-1}{10}}}$ times). ^{[7] : 166}

In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number ^[8] (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760,350,154,212,639,131).

There are sixty-one 3-uniform tilings.

Sixty-one is the exponent of the ninth Mersenne prime, ${\displaystyle M_{61}=2^{61}-1=2,305,843,009,213,693,951}$ ^[9] and the next candidate exponent for a potential fifth double Mersenne prime: ${\displaystyle M_{M_{61}}=2^{2305843009213693951}-1\approx 1.695\times 10^{694127911065419641}.}$ ^[10]

61 is also the largest prime factor in Descartes number, ^[11]

$${\displaystyle 3^{2}\times 7^{2}\times 11^{2}\times 13^{2}\times 19^{2}\times 61=198585576189.}$$

This number would be the only known odd perfect number if one of its composite factors (22021 = 19² × 61) were prime. ^[12]

61 is the largest prime number (less than the largest supersingular prime, 71) that does not divide the order of any sporadic group (including any of the pariahs).

The exotic sphere ${\displaystyle S^{61}}$ is the last odd-dimensional sphere to contain a unique smooth structure; ${\displaystyle S^{1}}$, ${\displaystyle S^{3}}$ and ${\displaystyle S^{5}}$ are the only other such spheres. ^{[13] [14]}

In science

• The chemical element with the atomic number 61 is promethium.

Astronomy

• Messier object M61, a magnitude 10.5 galaxy in the constellation Virgo
• The New General Catalogue object NGC 61, a double spiral galaxy in the constellation Cetus
• 61 Ursae Majoris is located about 31.1 light-years from the Sun.
• 61 Cygni was christened the "Flying Star" in 1792 by Giuseppe Piazzi (1746–1826) for its unusually large proper motion.

In other fields

See also: List of highways numbered 61

Sixty-one is:

• The number of the French department Orne
• The code for international direct dial phone calls to Australia
• 61* , a 2001 baseball movie directed by Billy Crystal
• Highway 61 Revisited is a Bob Dylan album
• The Highway 61 Blues Festival occurs annually in Leland, Mississippi
• Highway 61 is a 1991 film set on U.S. Route 61
• U.S. Route 61 is the highway that inspired so much attention on "Highway 61"
• Part 61 is a law created by the FAA regarding medical exams. This law has often come under attack by AOPA.
• The P-61 is the Northrop-designed fighter first designated as the XP-61. It first flew on May 26, 1942. It is also known as the Black Widow as it was the first fighter aircraft designed to be a night fighter
• Sixty 1 is a brand tobacco produced by Nationwide Tobacco
• 61A is the London address of Margot Wendice (Grace Kelly) and Tony Wendice (Ray Milland) in the movie Dial M for Murder
• 1 Liberty Place is one of Philadelphia's tallest buildings at 61 stories
• The number of cadets on The Summerall Guards
• The number of points required to win a "standard" game of cribbage ^[15]
• The maximum number of tables that can be joined in a single MariaDB or MySQL query ^[16]

In sports

• New York Yankees right fielder Roger Maris hit 61 home runs in 1961, breaking Babe Ruth's single-season record until it was surpassed in 1998 by Mark McGwire and Sammy Sosa. The American League record was broken 61 years later in 2022, by Aaron Judge.
• Nolan Ryan and Tom Seaver each had 61 career shutouts
• Hockey great Wayne Gretzky holds or shares 61 NHL records (40 for regular season, 15 for Stanley Cup playoff, and 6 for All-Star Games)
• Rotation, a variation of pool, is sometimes called 61
• Richie Evans' NASCAR Whelen Modified Tour car number was 61 until his death in 1985
• The number of the laps of the first Formula One night race, Singapore Grand Prix.

References

1. Sloane, N. J. A. (ed.). "SequenceA001844(Centered square numbers: a(n) is 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z equal to Y+1) ordered by increasing Z; then sequence gives Z values.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-09.
2. "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
3. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
4. "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
5. "Sloane's A063980 : Pillai primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
6. "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
7. Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.
8. "Sloane's A005235 : Fortunate numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
9. "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
10. "Mersenne Primes: History, Theorems and Lists". PrimePages . Retrieved 2023-10-22.
11. Holdener, Judy; Rachfal, Emily (2019). "Perfect and Deficient Perfect Numbers". The American Mathematical Monthly . 126 (6). Mathematical Association of America: 541–546. doi:10.1080/00029890.2019.1584515. MR   3956311. S2CID   191161070. Zbl   1477.11012 – via Taylor & Francis.
12. Sloane, N. J. A. (ed.). "SequenceA222262(Divisors of Descarte's 198585576189.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-27.
13. Wang, Guozhen; Xu, Zhouli (2017). "The triviality of the 61-stem in the stable homotopy groups of spheres". Annals of Mathematics . 186 (2): 501–580. arXiv: 1601.02184 . doi:10.4007/annals.2017.186.2.3. MR   3702672. S2CID   119147703.
14. Sloane, N. J. A. (ed.). "SequenceA001676(Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-10-22.
15. Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
16. MySQL Reference Manual – JOIN clause

• R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.