193 (number)

193 is the number of compositions of 14 into distinct parts.^[1] In decimal, it is the seventeenth full repetend prime, or long prime.^[2]

• It is the only odd prime ${\displaystyle p}$ known for which 2 is not a primitive root of ${\displaystyle 4p^{2}+1}$.^[3]

• It is the thirteenth Pierpont prime, which implies that a regular 193-gon can be constructed using a compass, straightedge, and angle trisector.^[4]

• It is part of the fourteenth pair of twin primes ${\displaystyle (191,193)}$,^[5] the seventh trio of prime triplets ${\displaystyle (193,197,199)}$,^[6] and the fourth set of prime quadruplets ${\displaystyle (191,193,197,199)}$.^[7]

Aside from itself, the friendly giant (the largest sporadic group) holds a total of 193 conjugacy classes.^[8] It also holds at least 44 maximal subgroups aside from the double cover of ${\displaystyle \mathbb {B} }$ (the forty-fourth prime number is 193).^[8][9][10]

193 is also the eighth numerator of convergents to Euler's number; correct to three decimal places: ${\displaystyle e\approx {\tfrac {193}{71}}\approx 2.718\;{\color {red}309\;859\;\ldots }}$ ^[11] The denominator is 71, which is the largest supersingular prime that uniquely divides the order of the friendly giant.^[12][13][14]

193 is also the current amount of member states that in the United Nations.^[15]

• 193