61 (number)

Template:For Template:Infobox number 61 (sixty-one) is the natural number following 60 and preceding 62.

In mathematics

File:Centered-hexagonal-61.png

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, <math>5^2 + 6^2</math>.<ref>Template:Cite OEIS</ref> It is also a centered decagonal number,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and a centered hexagonal number.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

61 is the fourth cuban prime of the form <math>p = \frac {x^{3} - y^{3}}{x - y}</math> where <math>x = y + 1</math>,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and the fourth Pillai prime since <math>8! + 1</math> is divisible by 61, but 61 is not one more than a multiple of 8.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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, ...<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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>p</math> which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length <math>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>\tfrac {p-1}{10}</math> times).<ref>Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.</ref>Template:Rp

In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> (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, <math>M_{61} = 2^{61} - 1 = 2,305,843,009,213,693,951</math><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and the next candidate exponent for a potential fifth double Mersenne prime: <math>M_{M_{61}} = 2^{2305843009213693951} - 1 \approx 1.695 \times 10^{694127911065419641}.</math><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

61 is also the largest prime factor in Descartes number,<ref>Template:Cite journal</ref>

<math display=block>3^2 \times 7^2 \times 11^2 \times 13^2 \times 19^2 \times 61 = 198585576189.</math>

This number would be the only known odd perfect number if one of its composite factors (22021 = 19² × 61) were prime.<ref>Template:Cite OEIS</ref>

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 <math>S^{61}</math> is the last odd-dimensional sphere to contain a unique smooth structure; <math>S^{1}</math>, <math>S^{3}</math> and <math>S^{5}</math> are the only other such spheres.<ref>Template:Cite journal</ref><ref>Template:Cite OEIS</ref>

References

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