137

137 is a natural number that lies between 136 and 138. It is a prime number and the 33rd prime in the sequence of primes. In common numeral systems, 137 is written as 137 in base 10, 10001001 in binary, and 0x89 in hexadecimal. The number is congruent to 1 modulo 4 and can be expressed as a sum of two squares: 137 = 11^2 + 4^2. As a prime, it is not a Sophie Germain prime, nor a safe prime.

In mathematics, 137 is notable for its role in various number-theoretic properties associated with primes of the form 4k + 1, and it can serve as an example in studies of representations of primes as sums of squares. The prime-counting function π(n) evaluated at 137 indicates that there are 33 primes less than or equal to 137.

In science, 137 has significance beyond pure numbers. The fine-structure constant alpha, a dimensionless quantity characterizing the strength of electromagnetic interactions, is approximately 1/137.036. Consequently, 137 is often cited in discussions of the inverse of this fundamental constant, though the exact value depends on the units used and is not equal to 137 precisely.

Cesium-137 is a well-known radioactive isotope with a half-life of about 30.17 years. It is produced in nuclear fission and decays by beta emission to barium-137m, which then decays to stable barium-137. Cesium-137 emits gamma radiation at 661.7 keV and has widespread applications in medical therapy, industrial radiography, and environmental dating.

Overall, 137 is a prime with several interesting mathematical representations and notable connections to physical constants and nuclear science.