Number reference
163 explained.
163: one hundred sixty-three, prime; factors, prime factorization, roots, binary, hexadecimal, and digit properties.
Identity and representations
- Written form
- one hundred sixty-three
- Classification
- prime
- Factors
- 1, 163
- Prime factorization
- 163
- Square root
- 12.767145
- Cube root
- 5.462556
- Binary
- 10100011
- Octal
- 243
- Hexadecimal
- A3
- Roman numeral
- CLXIII
- Digit sum
- 10
- Digital root
- 1
What makes 163 notable
- Prime. 163 is prime: it has exactly two divisors, 1 and itself, so it cannot be split into equal whole-number groups any other way.
163 in everyday use
One hundred sixty-three is the largest Heegner number, a distinction that makes it one of the most celebrated primes in higher mathematics. Because of it, the expression e raised to the power of pi times the square root of 163 comes within a trillionth of a whole number: 262,537,412,640,768,743.99999999999925. Charles Hermite noticed this in 1859, and Martin Gardner used it as an April Fools' hoax in 1975, claiming the value was exactly an integer. It is not, but the near miss is a genuine consequence of deep structure.
The same property explains a famous prime-generating formula. Euler's polynomial n² + n + 41 produces primes for every n from 0 to 39, and it works because 4 × 41 − 1 = 163. Number systems built on the square root of −163 have unique factorisation, just like ordinary integers, and 163 is the last number for which that happens. Beyond all this, 163 is simply prime, the 38th, and its digit sum is 10.
Nearby numbers
How to read this page
Because 163 is prime, the factor list stops at 1 and 163 itself, so it cannot be shared into equal whole-number groups. That is also what makes it useful as a building block: every larger number with 163 in its prime factorisation inherits it from here.
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