# 89 (number)
**89** (eighty-nine) is the natural number following 88 and preceding 90. It is a prime number, the 24th prime, and the 11th [Fibonacci](https://www.edgepedia.org/fibonacci) number, and it is the atomic number of the element actinium.[1](https://www.numberempire.com/89) • [2](https://mathworld.wolfram.com/FibonacciPrime.html) • [3](https://pubchem.ncbi.nlm.nih.gov/element/89)
| Key fact | Detail |
|---|---|
| Prime status | 89 is prime, the 24th prime; the previous prime is 83 and the next is 97, giving a prime gap of 8[1](https://www.numberempire.com/89) |
| Fibonacci | 89 is the 11th Fibonacci number (…, 34, 55, 89) and one of the Fibonacci primes[2](https://mathworld.wolfram.com/FibonacciPrime.html) |
| Sophie Germain prime | 2 × 89 + 1 = 179 is prime, so 89 is a Sophie Germain prime; 89 itself is not a safe prime[4](https://mathworld.wolfram.com/SophieGermainPrime.html) • [1](https://www.numberempire.com/89) |
| Arithmetic | Divisors are 1 and 89; Euler's totient φ(89) = 88; digit sum 17; digital root 8[1](https://www.numberempire.com/89) |
| Reciprocal | The decimal expansion of 1/89 generates the Fibonacci sequence, each term shifted one place further right until carries overlap[5](https://www.divinityatlas.com/numbers/89) |
| Chemistry | 89 is the atomic number of actinium (Ac), the element that gives its name to the actinide series[3](https://pubchem.ncbi.nlm.nih.gov/element/89) |
| Mersenne connection | 2¹¹ − 1 = 23 × 8911, so 89 divides the 11th Mersenne number[6](https://kar.kent.ac.uk/id/document/3468358) |
## Prime properties
As a prime, 89 has exactly two divisors, 1 and 89, and it cannot be shared into equal whole-number groups.[1](https://www.numberempire.com/89) • [7](https://numbertrove.com/number/89/) It sits in a gap of 8 between the primes 83 and 97, and the prime in position 89 is 461.[1](https://www.numberempire.com/89) It does not form a twin prime pair, because its neighbors are both composite: 87 = 3 × 29 and 91 = 7 × 13.[5](https://www.divinityatlas.com/numbers/89)
**Sophie Germain status.** A Sophie Germain prime is a prime p for which 2p + 1 is also prime; the resulting 2p + 1 is called a safe prime.[4](https://mathworld.wolfram.com/SophieGermainPrime.html) Since 2 × 89 + 1 = 179 is prime, 89 belongs to this class, appearing in the list of the first [Sophie Germain](https://www.edgepedia.org/sophie-germain) primes 2, 3, 5, 11, 23, 29, 41, 53, 83, 89.[4](https://mathworld.wolfram.com/SophieGermainPrime.html) • [8](https://numberworld.info/sophie-germain-primes) Sophie Germain primes matter in cryptography: choosing p as a Sophie Germain prime in Diffie-Hellman key exchange makes the safe prime q = 2p + 1 define a group whose [multiplicative order](https://www.edgepedia.org/multiplicative-order) is large and prime, making certain discrete logarithm attacks harder.[8](https://numberworld.info/sophie-germain-primes) Around 1825, Sophie Germain, the French mathematician, proved the first case of Fermat's last theorem for such primes.[4](https://mathworld.wolfram.com/SophieGermainPrime.html)
## Fibonacci connections
89 is the 11th term of the [Fibonacci sequence](https://www.edgepedia.org/fibonacci-sequence), following 55 and preceding 144.[5](https://www.divinityatlas.com/numbers/89) • [9](https://numberwise.app/number/89) A Fibonacci prime is a Fibonacci number that is also prime; the first few are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, corresponding to indices n = 3, 4, 5, 7, 11, 13, 17, 23, 29.[2](https://mathworld.wolfram.com/FibonacciPrime.html) Every Fibonacci number that is prime must have a prime index, with the exception of F₄ = 3, and 89's index 11 is indeed prime.[2](https://mathworld.wolfram.com/FibonacciPrime.html)
**The reciprocal 1/89.** One divided by 89 produces a decimal expansion in which the Fibonacci sequence appears, each term shifted one place further right, until the carries begin to overlap.[5](https://www.divinityatlas.com/numbers/89) The same expansion is discussed in the OEIS entry for the decimal expansion of 1/89, which also records primes of the form floor((1/89) × 10ⁿ) for n = 3, 5, and 631.[10](https://oeis.org/A021093)
**Mersenne and modular facts.** The 11th Mersenne number, 2¹¹ − 1 = 2047, factors as 23 × 89, so 89 appears as a prime factor precisely because 11, the index of 89 in the Fibonacci sequence, is also the Mersenne exponent involved.[6](https://kar.kent.ac.uk/id/document/3468358) In modular arithmetic, the OEIS sequence A020217 lists composite numbers n satisfying 89n−1 ≡ 1 (mod n), the base-89 analog of Poulet numbers, beginning 4, 8, 9, 15, 22, 33, 44, 45, 55, 85, 88, 99.[11](https://oeis.org/A020217)
## Basic arithmetic profile
The number 89 is deficient: the sum of its proper divisors is 1, so its deficiency, the amount by which the divisor sum falls short of the number itself, is 88.[12](https://numbermatics.com/n/89/) Its digit sum is 17 and its digital root is 8.[1](https://www.numberempire.com/89) It is not a [Lucas number](https://www.edgepedia.org/lucas-number), a [Bell number](https://www.edgepedia.org/bell-number), a [Catalan number](https://www.edgepedia.org/catalan-number), a highly composite number, or a polygonal number with s < 11.[1](https://www.numberempire.com/89)
## Actinium and other appearances
In chemistry, 89 is the atomic number of actinium, symbol Ac, meaning an actinium atom has 89 protons in its nucleus.[3](https://pubchem.ncbi.nlm.nih.gov/element/89) • [13](https://www.mathway.com/popular-problems/Chemistry/669575) [Actinium](https://www.edgepedia.org/actinium) opens the actinide series, the row of 15 metallic elements with atomic numbers 89 through 103 that sits below the main periodic table.[14](https://www.teachifyme.com/wp-content/uploads/2019/02/0620_w18_qp_33.pdf)
## References
1. [Properties of the number 89, Number Empire](https://www.numberempire.com/89)
2. [Fibonacci Prime, Wolfram MathWorld](https://mathworld.wolfram.com/FibonacciPrime.html)
3. [Actinium (Ac), Element 89, PubChem](https://pubchem.ncbi.nlm.nih.gov/element/89)
4. [Sophie Germain Prime, Wolfram MathWorld](https://mathworld.wolfram.com/SophieGermainPrime.html)
5. [89, Numbers, Divinity Atlas](https://www.divinityatlas.com/numbers/89)
6. [The Other Side of Mersenne, Kent Academic Repository](https://kar.kent.ac.uk/id/document/3468358)
7. [89: Number Facts, Factors & Representations, NumberTrove](https://numbertrove.com/number/89/)
8. [Sophie Germain Primes, NumberWorld](https://numberworld.info/sophie-germain-primes)
9. [89, Numberwise](https://numberwise.app/number/89)
10. [A021093, OEIS](https://oeis.org/A021093)
11. [A020217, OEIS](https://oeis.org/A020217)
12. [Number 89, Numbermatics](https://numbermatics.com/n/89/)
13. [Find the Number of Protons in Actinium, Mathway](https://www.mathway.com/popular-problems/Chemistry/669575)
14. [The Periodic Table of Elements, Cambridge IGCSE Chemistry paper](https://www.teachifyme.com/wp-content/uploads/2019/02/0620_w18_qp_33.pdf)
15. [Periodic Table of the Elements, Los Alamos National Laboratory](https://online.cctt.org/physicslab/content/phyapb/lessonnotes/dualnature/PeriodicWebSite.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Named individual integers*
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