← 60 61 62 →
Cardinalsixty-one
Ordinal61st
(sixty-first)
Factorizationprime
Prime18th
Divisors1, 61
Greek numeralΞΑ´
Roman numeralLXI
Binary1111012
Ternary20213
Senary1416
Octal758
Duodecimal5112

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, 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][a]

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

61 is also the largest prime factor in Descartes number,[17]

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

This number would be the only known odd perfect number if one of its composite factors (22021 = 192 × 61) were prime.[18]

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.[19][20]

Sixty-one is:

## 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, N. J. A. (ed.). "Sequence A001844 (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, N. J. A. (ed.). "Sequence A068599 (Number of n-uniform tilings.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
10. ^ a b Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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)#)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-07.
15. ^ "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
16. ^ "Mersenne Primes: History, Theorems and Lists". PrimePages. Retrieved 2023-10-22.
17. ^ 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.
18. ^ Sloane, N. J. A. (ed.). "Sequence A222262 (Divisors of Descarte's 198585576189.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
19. ^ 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.
20. ^ Sloane, N. J. A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-22.
21. ^ Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
22. ^ MySQL Reference Manual – JOIN clause
• R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.