# 193 (number)

193 (one hundred [and] ninety-three) is the natural number following 192 and preceding 194.

 ← 192 193 194 →
Cardinalone hundred ninety-three
Ordinal193rd
(one hundred ninety-third)
Factorizationprime
Prime44th
Divisors1, 193
Greek numeralΡϞΓ´
Roman numeralCXCIII
Binary110000012
Ternary210113
Senary5216
Octal3018
Duodecimal14112

## In mathematics

193 is a Pierpont prime number, implying that a 193-gon can be constructed using a compass, straightedge, and angle trisector.[1] It is the number of compositions of 14 into distinct parts.[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 part of the fourteenth pair of twin primes ${\displaystyle (191,193)}$ ,[4] the seventh trio of prime triplets ${\displaystyle (193,197,199)}$ ,[5] and the fourth set of prime quadruplets ${\displaystyle (191,193,197,199)}$ .[6]
• It is the tenth tribonacci number of the form ${\displaystyle a(n)=a(n-1)+a(n-2)+a(n-3)}$ , with ${\displaystyle a(0)=a(1)=a(2)=1}$ .[7]
• In decimal, it is the 17th full repetend prime, or long prime.[8]

Aside from itself, the friendly giant, the largest sporadic group, holds a total of 193 conjugacy classes.[9] It also holds at least 44 maximal subgroups (the forty-fourth prime number is 193).[9][10]

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

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A005109 (Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N. J. A. (ed.). "Sequence A032020 (Number of compositions (ordered partitions) of n into distinct parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-05-24.
3. ^ E. Friedman, "What's Special About This Number Archived 2018-02-23 at the Wayback Machine" Accessed 2 January 2006 and again 15 August 2007.
4. ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
5. ^ Sloane, N. J. A. (ed.). "Sequence A022005 (Initial members of prime triples (p, p+4, p+6).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
6. ^ Sloane, N. J. A. (ed.). "Sequence A136162 (List of prime quadruplets {p, p+2, p+6, p+8}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
7. ^ Sloane, N. J. A. (ed.). "Sequence A000213 (Tribonacci numbers...[with a0, a1 and a2 equal to 1])". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
8. ^ Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
9. ^ a b Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "ATLAS: Monster group M". ATLAS of Finite Group Representations.
10. ^ Wilson, Robert A. (2016). "Is the Suzuki group Sz(8) a subgroup of the Monster?" (PDF). Bulletin of the London Mathematical Society. 48 (2): 356. doi:10.1112/blms/bdw012. MR 3483073. S2CID 123219818.
11. ^ Sloane, N. J. A. (ed.). "Sequence A007676 (Numerators of convergents to e.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
12. ^ Sloane, N. J. A. (ed.). "Sequence A007677 (Denominators of convergents to e.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
13. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes: primes dividing order of Monster simple group.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-03-02.
14. ^ Luis J. Boya (2011-01-16). "Introduction to Sporadic Groups". Symmetry, Integrability and Geometry: Methods and Applications. 7: 13. arXiv:1101.3055. Bibcode:2011SIGMA...7..009B. doi:10.3842/SIGMA.2011.009. S2CID 16584404.