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, 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 as a centered hexagonal number

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]}

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.

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