57 (number)
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|---|---|---|---|---|
| Cardinal | fifty-seven | |||
| Ordinal | 57th (fifty-seventh) | |||
| Factorization | 3 × 19 | |||
| Divisors | 1, 3, 19, 57 | |||
| Greek numeral | ΝΖ´ | |||
| Roman numeral | LVII, lvii | |||
| Binary | 1110012 | |||
| Ternary | 20103 | |||
| Senary | 1336 | |||
| Octal | 718 | |||
| Duodecimal | 4912 | |||
| Hexadecimal | 3916 | |||
57 (fifty-seven) is the natural number following 56 and preceding 58. It is a composite number.
In mathematics
[edit]57 has prime factorization , and is therefore a semiprime.[1] Its proper divisors are 1, 3, and 19, whose sum is 23, so 57 is a deficient number. Since both prime factors are congruent to 3 modulo 4, is a Blum integer.[2] It is a Leyland number, because[3] .
The split Lie algebra E7+1/2 has a 57-dimensional Heisenberg algebra as its nilradical, and the smallest possible homogeneous space for E8 is also 57-dimensional.[4]
Although fifty-seven is not prime, it is jokingly known as the Grothendieck prime after a legend in which the mathematician Alexander Grothendieck gave it as an example of a prime number, not realizing it was divisible by three and nineteen.[5] The same error was made by another famous mathematician, Hermann Weyl, in a published article.[6]
References
[edit]- ↑ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A016105 (Blum integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Vogan, David (2007), "The character table for E8" (PDF), Notices of the American Mathematical Society, 54 (9): 1122–1134, MR 2349532
- ↑ Jackson, Allyn (2004b). "Comme Appelé du Néant—As if Summoned from the Void: The Life of Alexandre Grothendieck" (PDF). Notices of the American Mathematical Society. 51 (10). Providence, RI: American Mathematical Society: 1196, 1197. MR 2104915. Zbl 1168.01339.
- ↑ Weyl, Hermann (1951). "A Half-Century of Mathematics". American Mathematical Monthly. 58 (5). Washington, D.C.: Mathematical Association of America: 532. doi:10.1080/00029890.1951.11999734. JSTOR 2306319. S2CID 126101329.
An old conjecture of Goldbach's maintains that there even come along again and again pairs of primes of the smallest possible difference 2, like 57 and 59.
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