137 (number)
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137 is a natural number that follows 136 and precedes 138. It is a prime number, meaning that only 1 and 137 divide it evenly. It is also odd, since it is not divisible by 2.
In decimal notation, 137 consists of three digits: 1, 3, and 7. The sum of its digits is 11. Among prime numbers less than 200, 137 is followed by 139.
Mathematical properties
137 is a prime number of the form 4k+1 (137 = 4 × 34 + 1). According to Fermat's theorem on primes of this form, it can be expressed as a sum of two squares: 137 = 16 + 121, or 4² + 11². This representation is unique for primes of the form 4k+1.
137 is also one of a pair of twin primes: 139 is another prime number exactly two units larger, so 137 and 139 are twin primes. When squared, 137² = 18769.
Significance in physics
137 is notable in physics because of the fine-structure constant (α), which is approximately equal to 1/137. This constant measures the strength of the electromagnetic interaction. The exact value is not precisely 137; it is approximately 1/137.036, so α is not exactly a rational number.
During the 20th century, several physicists, including theoretical physicists, attempted to derive 137 as a fundamental number from pure mathematics. These attempts were not accepted by the modern physics community; today α is understood as an empirical constant that cannot be derived from mathematics alone.