61 (number) explained

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,

.^[1] It is also a centered decagonal number,^[2] and a centered hexagonal number.^[3]

61 is the fourth cuban prime of the form

where ,^[4] and the fourth Pillai prime since 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

which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, times).^[7]

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,

^[9] and the next candidate exponent for a potential fifth double Mersenne prime: ^[10]

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

$$3^2\; \backslash times\; 7^2\; \backslash times\; 11^2\; \backslash times\; 13^2\; \backslash times\; 19^2\; \backslash times\; 61\; =\; 198585576189.$$

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).

is the last odd-dimensional sphere to contain a unique smooth structure; , and are the only other such spheres.^{[13] [14]}

References

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

Notes and References

1. 2024-02-09 .
2. Web site: Sloane's A062786 : Centered 10-gonal numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
3. Web site: Sloane's A003215 : Hex (or centered hexagonal) numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
4. Web site: Sloane's A002407 : Cuban primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
5. Web site: Sloane's A063980 : Pillai primes. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
6. Web site: Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
7. Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.
8. Web site: Sloane's A005235 : Fortunate numbers. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
9. Web site: Sloane's A000043 : Mersenne exponents. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-05-30.
10. Web site: Mersenne Primes: History, Theorems and Lists . . 2023-10-22 .
11. Holdener . Judy . Rachfal . Emily . Perfect and Deficient Perfect Numbers . . 126 . 6 . . 2019 . 541–546 . 10.1080/00029890.2019.1584515 . 3956311 . 1477.11012 . 191161070 . . subscription .
12. 2024-02-27 .
13. Wang . Guozhen . Xu . Zhouli . The triviality of the 61-stem in the stable homotopy groups of spheres . . 186 . 2017 . 2 . 501–580 . 10.4007/annals.2017.186.2.3 . 3702672 . 1601.02184 . 119147703 .
14. 2023-10-22 .