61 (number)

From HandWiki
Short description: Natural number
← 60 61 62 →
Cardinalsixty-one
Ordinal61st
(sixty-first)
Factorizationprime
Prime18th
Divisors1, 61
Greek numeralΞΑ´
Roman numeralLXI
Binary1111012
Ternary20213
Quaternary3314
Quinary2215
Senary1416
Octal758
Duodecimal5112
Hexadecimal3D16
Vigesimal3120
Base 361P36

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. It is the sum of two consecutive squares, [math]\displaystyle{ 5^2 + 6^2. }[/math] It is also a centered decagonal number,[1] a centered hexagonal number,[2] and a centered square number.[3]

61 is the fourth cuban prime of the form [math]\displaystyle{ p = \frac {x^{3} - y^{3}}{x - y} }[/math] where [math]\displaystyle{ x = y + 1 }[/math],[4] and the forth Pillai prime since [math]\displaystyle{ 8! + 1 }[/math] 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 [math]\displaystyle{ p }[/math] which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length [math]\displaystyle{ p - 1, }[/math] where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, [math]\displaystyle{ \tfrac {p-1}{10} }[/math] 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, where on the other hand, there are one hundred and fifty-one 4-uniform tilings[9] (with 61 the eighteenth prime number, and 151 the thirty-sixth, twice the index value).[10][lower-alpha 1]

61 is the exponent of the ninth Mersenne prime, [math]\displaystyle{ M_{61} = 2^{61} - 1 = 2,305,843,009,213,693,951 }[/math][15] and the next candidate exponent for a potential fifth double Mersenne prime: [math]\displaystyle{ M_{M_{61}} = 2^{2305843009213693951} - 1 \approx 1.695 \times 10^{694127911065419641}. }[/math][16]

The exotic sphere [math]\displaystyle{ S^{61} }[/math] is the last odd-dimensional sphere to contain a unique smooth structure; [math]\displaystyle{ S^{1} }[/math], [math]\displaystyle{ S^{3} }[/math] and [math]\displaystyle{ S^{5} }[/math] are the only other such spheres.[17][18]

In science

Astronomy

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[19]
  • The maximum number of tables that can be joined in a single MariaDB or MySQL query[20]

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.

Notelist

  1. ↑ Otherwise, there are eleven total 1-uniform tilings (the regular and semiregular tilings), and twenty 2-uniform tilings (where 20 is the eleventh composite number;[11] together these values add to 31, the eleventh prime).[10][12] The sum of the first twenty integers is the fourth primorial 210,[13][14] equal to the product of the first four prime numbers, and 1, whose collective sum generated is 18.

References

  1. ↑ "Sloane's A062786 : Centered 10-gonal numbers". OEIS Foundation. https://oeis.org/A062786. 
  2. ↑ "Sloane's A003215 : Hex (or centered hexagonal) numbers". OEIS Foundation. https://oeis.org/A003215. 
  3. ↑ "Sloane's A001844 : Centered square numbers". OEIS Foundation. https://oeis.org/A001844. 
  4. ↑ "Sloane's A002407 : Cuban primes". OEIS Foundation. https://oeis.org/A002407. 
  5. ↑ "Sloane's A063980 : Pillai primes". OEIS Foundation. https://oeis.org/A063980. 
  6. ↑ "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". OEIS Foundation. https://oeis.org/A007629. 
  7. ↑ Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.
  8. ↑ "Sloane's A005235 : Fortunate numbers". OEIS Foundation. https://oeis.org/A005235. 
  9. ↑ Sloane, N. J. A., ed. "Sequence A068599 (Number of n-uniform tilings.)". OEIS Foundation. https://oeis.org/A068599. Retrieved 2024-01-07. 
  10. ↑ 10.0 10.1 Sloane, N. J. A., ed. "Sequence A000040 (The prime numbers.)". OEIS Foundation. https://oeis.org/A000040. Retrieved 2024-01-07. 
  11. ↑ Sloane, N. J. A., ed. "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". OEIS Foundation. https://oeis.org/A002808. Retrieved 2024-01-07. 
  12. ↑ Sloane, N. J. A., ed. "Sequence A299782 (a(n) is the total number of k-uniform tilings, for k equal to 1..n.)". OEIS Foundation. https://oeis.org/A299782. Retrieved 2024-01-07. 
  13. ↑ Sloane, N. J. A., ed. "Sequence A000217 (Triangular numbers: a(n) is the binomial(n+1,2) equal to n*(n+1)/2 or 0 + 1 + 2 + ... + n.)". OEIS Foundation. https://oeis.org/A000217. Retrieved 2024-01-07. 
  14. ↑ Sloane, N. J. A., ed. "Sequence A002110 (Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#)". OEIS Foundation. https://oeis.org/A002110. Retrieved 2024-01-07. 
  15. ↑ "Sloane's A000043 : Mersenne exponents". OEIS Foundation. https://oeis.org/A000043. 
  16. ↑ "Mersenne Primes: History, Theorems and Lists". https://t5k.org/mersenne/index.html#unknown. 
  17. ↑ 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. doi:10.4007/annals.2017.186.2.3. 
  18. ↑ Sloane, N. J. A., ed. "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". OEIS Foundation. https://oeis.org/A001676. Retrieved 2023-10-22. 
  19. ↑ Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
  20. ↑ MySQL Reference Manual – JOIN clause
  • R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.

External links



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