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Natural number
61 (sixty-one) is the natural number following 60 and preceding 62.
61 as a centered hexagonal number61 is the 18th prime number, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, 5 2 + 6 2 {\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 p = x 3 − y 3 x − y {\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}} where x = y + 1 {\displaystyle x=y+1} ,[4] and the fourth Pillai prime since 8 ! + 1 {\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 p {\displaystyle p} which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length p − 1 , {\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, p − 1 10 {\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, M 61 = 2 61 − 1 = 2 , 305 , 843 , 009 , 213 , 693 , 951 {\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: M M 61 = 2 2305843009213693951 − 1 ≈ 1.695 × 10 694127911065419641 . {\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]
3 2 × 7 2 × 11 2 × 13 2 × 19 2 × 61 = 198585576189. {\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 S 61 {\displaystyle S^{61}} is the last odd-dimensional sphere to contain a unique smooth structure; S 1 {\displaystyle S^{1}} , S 3 {\displaystyle S^{3}} and S 5 {\displaystyle S^{5}} are the only other such spheres.[13][14]
In Fifteen-Ball disciplines of pool, 61 points are required to win a rack. Since each ball is worth its respective printed number of points, the total points are 1 + 2 + 3 + … + 14 + 15 = 120, and therefore 61/120 points ensures a win.
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