91 (number)

91 (ninety-one) is the natural number following 90 and preceding 92.

← 90 91 92 →
← 90 91 92 93 94 95 96 97 98 99 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	ninety-one
Ordinal	91st (ninety-first)
Factorization	7 × 13
Divisors	1, 7, 13, 91
Greek numeral	ϞΑ´
Roman numeral	XCI, xci
Binary	10110112
Ternary	101013
Senary	2316
Octal	1338
Duodecimal	7712
Hexadecimal	5B16

In mathematics

91 is a square pyramidal number.

91 is a centered hexagonal number.

91 as the sum and difference of two positive cubes

91 is:

• the twenty-seventh distinct semiprime^[1] and the second of the form (7.q), where q is a higher prime.
• the aliquot sum of 91 is 21; itself a semiprime, within an aliquot sequence of two composite numbers (91, 21, 11, 1, 0) to the prime in the 11-aliquot tree. 91 is the fourth composite number in the 11-aliquot tree. (91, 51, 21, 18).
• the 13th triangular number.^[2]
• a hexagonal number,^[3] one of the few such numbers to also be a centered hexagonal number.^[4]
• a centered nonagonal number.^[5]
• a centered cube number.^[6]
• a square pyramidal number, being the sum of the squares of the first six integers.^[7]
• the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes):^[8]
  91 = 6³ + (−5)³ = 4³ + 3³. (See 1729 for more details).
  This implies that 91 is the second cabtaxi number.
• the smallest positive integer expressible as a sum of six distinct squares:
  91 = 1² + 2² + 3² + 4² + 5² + 6².
• The only other ways to write 91 as a sum of distinct squares are:
  91 = 1² + 4² + 5² + 7² and
• 91 = 1² + 3² + 9².
• the smallest pseudoprime satisfying the congruence 3ⁿ ≡ 3 mod n.^[9]
• a repdigit in base 9 (111₉).
• palindromic in bases 3 (10101₃), 9 (111₉), and 12 (77₁₂).
• a Riordan number.^[10]
• the smallest number that looks prime but is not, proven using the Rotten Theorem by John Conway.^[11][12]

The decimal equivalent of the fraction 1⁄91 can be obtained by using powers of 9.

In science

• McCarthy 91 function, a recursive function in discrete mathematics