65,537
65,537 is the public exponent used by most RSA encryption in the world.
- Odd
- Prime
Where 65,537 fits
Why 65,537 is special
It is chosen because it is prime, is 2^16 + 1, and has only two bits set — so encryption is fast — while being large enough to avoid the attacks that broke the earlier choice of 3. It is also the largest known Fermat prime, and probably the last.
65,537 is part of
Factors & digit properties
- Divisors (2)
- 1, 65537
- Prime factorisation
- 65537
- Digit sum
- 26
- Digital root
- 8
Representations
- Decimal
- 65537
- Scientific notation
- 6.554 × 10^4
Roman numerals
Roman numerals as modelled here goes up to 3,999.
Binary
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 × 65,536 + 0 × 32,768 + 0 × 16,384 + 0 × 8,192 + 0 × 4,096 + 0 × 2,048 + 0 × 1,024 + 0 × 512 + 0 × 256 + 0 × 128 + 0 × 64 + 0 × 32 + 0 × 16 + 0 × 8 + 0 × 4 + 0 × 2 + 1 × 1 = 65,537
Octal
2 0 0 0 0 1
2 × 32,768 + 0 × 4,096 + 0 × 512 + 0 × 64 + 0 × 8 + 1 × 1 = 65,537
Hexadecimal
1 0 0 0 1
1 × 65,536 + 0 × 4,096 + 0 × 256 + 0 × 16 + 1 × 1 = 65,537
Read as powers of two: 65,536 + 1 = 65,537.
In historical numeral systems
Egyptian
6 fingers, 5 lotuses, 5 coils of rope, 3 hobbles, 7 strokes
6 × 10,000 + 5 × 1,000 + 5 × 100 + 3 × 10 + 7 × 1 = 65,537. The signs are simply repeated and added; where they sit on the line is a matter of layout, not value.
Systems that simply cannot reach this value are left out. Where a script needs a font most readers do not have, the reading is given instead of the glyphs — see number systems for each system in full.
Other mathematical patterns
- Deficient number
- Twin prime
Related numbers & connections
Two of the five known Fermat primes. 65,537 is the largest, and is the public exponent used by most RSA encryption in the world because its bit pattern makes the arithmetic cheap.
View 257 →Shares a distinctive class
- is associated with257
why?
Strength 0.50, confidence "high".
Technology & computing
65,537 is chosen as the RSA public exponent because it is prime, is 2^16 + 1, and has only two bits set, so encryption is fast — while being large enough to avoid the attacks that made the earlier choice of 3 unsafe. It is also the largest known Fermat prime.
DocumentedTechnology & computing
Mathematics
65,537 is 2^(2^4) + 1, the fifth and largest known Fermat prime. Fermat conjectured every number of that form was prime; Euler found a factor of the next one in 1732, and no sixth has been found since. Gauss showed a regular polygon can be constructed with compass and straightedge exactly when its number of sides is a power of two times distinct Fermat primes — so a regular 65,537-sided polygon is constructible, and Johann Hermes spent about a decade writing out the construction in the 1890s.
DocumentedMathematicsPeriod: 1732 onwards
Common questions
Is 65,537 a prime number?
Yes, 65,537 is prime — it has exactly two positive divisors: 1 and itself.
What are the factors of 65,537?
The factors of 65,537 are 1, 65537.
About this page's data
Mathematical facts:computed directly from the number's value — nothing to source or verify externally.
Contextual records: 2 reviewer-confirmed records, sourced and reviewed before publication.
Last reviewed: 15 July 2026