# 91 (number)

 ← 90 91 92 →
Cardinal ninety-one
Ordinal 91st
(ninety-first)
Factorization 7 × 13
Divisors 1, 7, 13, 91
Roman numeral XCI
Binary 10110112
Ternary 101013
Quaternary 11234
Quinary 3315
Senary 2316
Octal 1338
Duodecimal 7712
Vigesimal 4B20
Base 36 2J36

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

## In mathematics

91 is:

• the twenty-seventh distinct semiprime and the second of the form (7.q).
• a triangular number.
• a hexagonal number,[1] one of the few such numbers to also be a centered hexagonal number.[2]
• a centered nonagonal number.[3]
• a centered cube number.[4]
• a square pyramidal number, being the sum of the squares of the first six integers.[5]
• 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): 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.[6]
• a repdigit in base 9 (1119).
• palindromic in bases 3 (101013), 9, and 12 (7712).
• the fourth composite number in the 11-aliquot tree. The aliquot sum of 91 is 21 within the aliquot sequence (91,21,11,1,0).

The decimal equivalent of the fraction 191 can be obtained by using powers of 9.

## In science

In astronomy,

Messier object M91, a magnitude 11.5 spiral galaxy in the constellation Coma Berenices
The New General Catalogue object NGC 91, a single star in the constellation Andromeda
The Saros number of the solar eclipse series which began on August 6, 159 BC and ended on 1175 October. The duration of Saros series 91 was 1334.2 years, and it contained 75 solar eclipses.
The Saros number of the lunar eclipse series which began on 161 May and ended on 1459 July. The duration of Saros series 91 was 1298.1 years, and it contained 73 lunar eclipses.

## In other fields

Ninety-one is also:

## References

1. ^ "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
3. ^ "Sloane's A060544 : Centered 9-gonal (also known as nonagonal or enneagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
4. ^ "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
5. ^ "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.