Number analysis
65,537
The largest known Fermat prime, and a common RSA public exponent.
The largest known Fermat prime, and a common RSA public exponent.
The number itself
- Value
- 65,537
- Digits
- 5
- Digit sum
- 26
- Reversed
- 73,556
- Palindrome
- No
6 + 5 + 5 + 3 + 7
Digit sum repeated until one digit remains; equals n mod 9 (with 9 for multiples of 9).
Reverses to 73,556.
Parity and primality
- Parity
- Odd
- Prime
- Yes
- Prime factorisation
- 65537
- Squarefree
- Yes
- Möbius μ(n)
- -1
- Fermat prime
- 2^16 + 1
- Twin prime
- Yes — paired with 65,539
- Nearest primes
- 65,521 ← → 65,539
Leaves remainder 1 on division by 2.
Divisible only by 1 and itself.
1 distinct prime factor, 1 with multiplicity.
No prime divides it twice.
(−1) to the power of 1 distinct prime factors.
A gap of 18 between them.
Divisors
- Sum of divisors σ(n)
- 65,538
- Aliquot sum
- 1
- Classification
- Deficient
- Euler totient φ(n)
- 65,536
1, 65,537
The proper divisors — everything except n itself.
Its proper divisors fall short by 65,536.
How many integers from 1 to n share no factor with n.
Sums and additive structure
- Sum of two squares
- Yes
- Sum of three squares
- Yes
- Sum of four squares
- Yes
1² + 256² = 65,537
Legendre's three-square theorem allows it.
Lagrange's four-square theorem: every non-negative integer is, without exception.
Shapes and sequences
- Figurate shapes
- None
Not triangular, square, pentagonal, hexagonal, cubic, Fibonacci, Lucas, Catalan, factorial or a power of two.
Digit curiosities
- Happy number
- No
- Harshad number
- No
- Collatz trajectory
- 99 steps to 1
Summing the squares of its digits falls into the 4 → 16 → 37 → … → 4 cycle.
Not divisible by its digit sum 26.
Peaks at 196,612 along the way.
Written other ways
- Binary
- 10000000000000001
- Octal
- 200001
- Hexadecimal
- 10001
- Base 36
- 1EKH
- In words
- sixty-five thousand five hundred and thirty-seven
- Scientific notation
- 6.5537 × 10^4
Standard Roman numerals only run from 1 to 3,999.
Is 65,537 a prime number?
Yes. 65,537 has no divisors other than 1 and itself.
Nearby numbers
References
Further reading: Wikipedia · Wikidata
Notations, naming and the links above come from Wikidata, whose structured data is dedicated to the public domain under CC0-1.0. Retrieved 2026-10-07.