# 43 (number)

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

Cardinal | forty-three | |||

Ordinal | 43rd (forty-third) | |||

Factorization | prime | |||

Prime | 14th | |||

Divisors | 1, 43 | |||

Greek numeral | ΜΓ´ | |||

Roman numeral | XLIII | |||

Binary | 101011_{2} | |||

Ternary | 1121_{3} | |||

Quaternary | 223_{4} | |||

Quinary | 133_{5} | |||

Senary | 111_{6} | |||

Octal | 53_{8} | |||

Duodecimal | 37_{12} | |||

Hexadecimal | 2B_{16} | |||

Vigesimal | 23_{20} | |||

Base 36 | 17_{36} |

**43** (**forty-three**) is the natural number following 42 and preceding 44.

## Mathematics

Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is 47. 43 is the smallest prime that is not a Chen prime. It is also the third Wagstaff prime.^{[1]}

43 is the fourth term of Sylvester's sequence, one more than the product of the previous terms (2 × 3 × 7).^{[2]}

43 is a centered heptagonal number.^{[3]}

Let *a*_{0} = *a*_{1} = 1, and thenceforth *a*_{n} = 1/*n* − 1(*a*_{0}^{2} + *a*_{1}^{2} + ... + *a*_{n − 1}^{2}). This sequence continues 1, 1, 2, 3, 5, 10, 28, 154... (sequence A003504 in the OEIS). *a*_{43} is the first term of this sequence that is not an integer.

43 is a Heegner number.^{[4]}

43 is the largest prime which divides the order of the Janko group J_{4}.

43 is a repdigit in base 6 (111).

43 is the largest natural number that is not an (original) McNugget number.^{[5]}

43 is the smallest prime number expressible as the sum of 2, 3, 4, or 5 different primes:

- 43 = 41 + 2
- 43 = 11 + 13 + 19
- 43 = 2 + 11 + 13 + 17
- 43 = 3 + 5 + 7 + 11 + 17.

43 is the smallest number with the property 43 = 4*prime(4) + 3*prime(3). Where prime(n) is the n-th prime number. There are only two numbers with that property, the other one is 127.

When taking the first six terms of the Taylor series for computing e, one obtains

- [math]\displaystyle{ \sum_{i=0}^{5}\frac{1}{i!}=\frac{163}{60}=2+\frac{43}{60},\ }[/math]

which is also five minus the fifth harmonic number.

Every solvable configuration of the Fifteen puzzle can be solved in no more than 43 multi-tile moves (i.e. when moving two or three tiles at once is counted as one move).^{[6]}

## In other fields

### Science

The chemical element with the atomic number 43 is technetium. It has the lowest atomic number of any element that does not possess stable isotopes.

### Music

The number of notes in Harry Partch's 43-tone scale of just intonation.

### Literature

"Number 43", in *Sonnets from the Portuguese* (1850), is one of Elizabeth Barrett Browning's most famous poems.

### Mysticism

43 is the number of triangles inside the Sri Yantra.

## Notes

- ↑ Sloane, N. J. A., ed. "Sequence A000979 (Wagstaff primes)". OEIS Foundation. https://oeis.org/A000979. Retrieved 2016-05-30.
- ↑ Sloane, N. J. A., ed. "Sequence A000058 (Sylvester's sequence)". OEIS Foundation. https://oeis.org/A000058. Retrieved 2016-05-30.
- ↑ Sloane, N. J. A., ed. "Sequence A069099 (Centered heptagonal numbers)". OEIS Foundation. https://oeis.org/A069099. Retrieved 2016-05-30.
- ↑ Sloane, N. J. A., ed. "Sequence A003173 (Heegner numbers)". OEIS Foundation. https://oeis.org/A003173. Retrieved 2016-05-30.
- ↑ Sloane, N. J. A., ed. "Sequence A065003 (Not McNugget numbers)". OEIS Foundation. https://oeis.org/A065003. Retrieved 2016-05-30.
- ↑ "The Fifteen Puzzle can be solved in 43 "moves"". Domain of the Cube Forum

## References

- Lehmer, Derrick,
*List of prime numbers from 1 to 10,006,721*, Carnegie Institution of Washington, 1914 - Wells, David,
*Prime Numbers: The Most Mysterious Figures in Math*, Wiley, 2005, ISBN:0-471-46234-9 - Crandall, Richard and Pomerance, Carl,
*Prime Numbers: A Computational Perspective*, Springer, 2005, ISBN:0-387-25282-7

### Further reading

Lenstra, Hendrik (2009). *Ode to the number 43* (In Dutch). Nieuw Arch. Wiskd. Amsterdam, NL: Koninklijk Wiskundig Genootschap (5) 10, No. 4: 240-244. MR2590266 Zbl 1263.00002

Original source: https://en.wikipedia.org/wiki/43 (number).
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