# 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...

## In science

The chemical element with the atomic number 61 (promethium), a lanthanide, is the element with the secondary lowest ordinal number that does not possess any stable isotopes. The promethium preceding element with atomic number 60 (neodymium) and the promethium following element 62 (samarium) have all stable isotopes.

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