Number reference
137 explained.
137: one hundred thirty-seven, prime; factors, prime factorization, roots, binary, hexadecimal, and digit properties.
Identity and representations
- Written form
- one hundred thirty-seven
- Classification
- prime
- Factors
- 1, 137
- Prime factorization
- 137
- Square root
- 11.7047
- Cube root
- 5.155137
- Binary
- 10001001
- Octal
- 211
- Hexadecimal
- 89
- Roman numeral
- CXXXVII
- Digit sum
- 11
- Digital root
- 2
What makes 137 notable
- Prime. 137 is prime: it has exactly two divisors, 1 and itself, so it cannot be split into equal whole-number groups any other way.
137 in everyday use
One hundred thirty-seven is the physicists' number. The fine-structure constant, which sets the strength of the electromagnetic force, is very nearly 1/137 — the measured value is about 1/137.036. Because it is a pure number with no units, its value seems to demand an explanation, and none has been found. Wolfgang Pauli was preoccupied with it and, by a coincidence often retold, died in hospital room 137. Arthur Eddington tried to derive exactly 137 from first principles and was proven wrong when measurements sharpened. Richard Feynman called it one of the greatest mysteries of physics.
Arithmetically, 137 is the 33rd prime and the smaller member of the twin-prime pair 137 and 139. It is a Pythagorean prime, one that leaves remainder 1 when divided by 4, which guarantees it can be written as a sum of two squares: 11² + 4² = 121 + 16. Its digits add to 11, also prime, and reversing them gives 731, which factors as 17 × 43.
Nearby numbers
How to read this page
Because 137 is prime, the factor list stops at 1 and 137 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 137 in its prime factorisation inherits it from here.
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