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

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, ${\displaystyle 5^{2}+6^{2}}$. It is also a centered decagonal number, and a centered hexagonal number.

61 is the fourth cuban prime of the form ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$ where ${\displaystyle x=y+1}$, and the fourth Pillai prime since ${\displaystyle 8!+1}$ is divisible by 61, but 61 is not one more than a multiple of 8. 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, ...

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).

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