# 61 (number)

For the Cyrillic letter, see Ы.
 ← 60 61 62 →
Cardinal sixty-one
Ordinal 61st
(sixty-first)
Factorization prime
Divisors 1, 61
Roman numeral LXI
Binary 1111012
Ternary 20213
Quaternary 3314
Quinary 2215
Senary 1416
Octal 758
Duodecimal 5112
Vigesimal 3120
Base 36 1P36

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

## In mathematics

It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form $p = (x^3 - y^3) / (x - y), x = y + 1$.

Sixty-one is the smallest proper prime, a prime p which ends in the digit 1in base 10 and whose reciprocal in base 10 has a repeating sequence with length p-1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, (p-1)/10 times).[1]:166

61 is 9th Mersenne prime exponent. (261 − 1 = 2,305,843,009,213,693,951)

Sixty-one is the sum of two squares, 52 + 62, and it is also a centered square number, a centered hexagonal number and a centered decagonal number.

It is the sixth Euler zigzag number (or Up/down number).

In base 14, 61 is a unique prime since no other prime has a 6-digit period in base 14.

Since 8! + 1 is divisible by 61 but 61 is not one more than a multiple of 8, 61 is a Pillai prime. 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).

It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61...

Sixty-one is:

## References

• R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.
1. ^ Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952.
2. ^ Hoyle, Edmund Hoyle's Official Rules of Card Games pub. Gary Allen Pty Ltd, (2004) p. 470
3. ^ MySQL Reference Manual – Limits of Joins