1105 (number)
| ||||
|---|---|---|---|---|
| Cardinal | one thousand one hundred five | |||
| Ordinal | 1105th (one thousand one hundred fifth) | |||
| Factorization | 5 × 13 × 17 | |||
| Greek numeral | ,ΑΡΕ´ | |||
| Roman numeral | MCV | |||
| Binary | 100010100012 | |||
| Ternary | 11112213 | |||
| Quaternary | 1011014 | |||
| Quinary | 134105 | |||
| Senary | 50416 | |||
| Octal | 21218 | |||
| Duodecimal | 78112 | |||
| Hexadecimal | 45116 | |||
| Vigesimal | 2F520 | |||
| Base 36 | UP36 | |||
1105 (eleven hundred [and] five, or one thousand one hundred [and] five) is the natural number following 1104 and preceding 1106.
1105 is the smallest positive integer that is a sum of two positive squares in exactly four different ways,[1][2] a property that can be connected (via the sum of two squares theorem) to its factorization 5 × 13 × 17 as the product of the three smallest prime numbers that are congruent to 1 modulo 4.[2][3] It is also the second-smallest Carmichael number, after 561,[4][5] one of the first four Carmichael numbers identified by R. D. Carmichael in his 1910 paper introducing this concept.[5][6]
Its binary representation 10001010001 and its base-4 representation 101101 are both palindromes,[7] and (because the binary representation has nonzeros only in even positions and its base-4 representation uses only the digits 0 and 1) it is a member of the Moser–de Bruijn sequence of sums of distinct powers of four.[8]
As a number of the form [math]\displaystyle{ \tfrac{n(n^2+1)}{2} }[/math] for [math]\displaystyle{ n={} }[/math]13, 1105 is the magic constant for 13 × 13 magic squares,[9] and as a difference of two consecutive fourth powers (1105 = 74 − 64)[10][11] it is a rhombic dodecahedral number (a type of figurate number), and a magic number for body-centered cubic crystals.[10][12] These properties are closely related: the difference of two consecutive fourth powers is always a magic constant for an odd magic square whose size is the sum of the two consecutive numbers (here 7 + 6 = 13).[10]
References
- ↑ Sloane, N. J. A., ed. "Sequence A016032 (Least positive integer that is the sum of two squares of positive integers in exactly n ways)". OEIS Foundation. https://oeis.org/A016032.
- ↑ 2.0 2.1 Tenenbaum, Gérald (1997). "1105: first steps in a mysterious quest". in Graham, Ronald L.; Nešetřil, Jaroslav. The mathematics of Paul Erdős, I. Algorithms and Combinatorics. 13. Berlin: Springer. pp. 268–275. doi:10.1007/978-3-642-60408-9_21.
- ↑ Sloane, N. J. A., ed. "Sequence A006278 (product of the first n primes congruent to 1 (mod 4))". OEIS Foundation. https://oeis.org/A006278.
- ↑ Sloane, N. J. A., ed. "Sequence A002997 (Carmichael numbers)". OEIS Foundation. https://oeis.org/A002997.
- ↑ 5.0 5.1 Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 9. Springer-Verlag, New York. p. 136. doi:10.1007/978-0-387-21850-2. ISBN 0-387-95332-9. https://books.google.com/books?id=hgfSBwAAQBAJ&pg=PA136.
- ↑ Carmichael, R. D. (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society 16 (5): 232–238. doi:10.1090/S0002-9904-1910-01892-9. https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-16/issue-5/Note-on-a-new-number-theory-function/bams/1183420617.full.
- ↑ Sloane, N. J. A., ed. "Sequence A097856 (Numbers that are palindromic in bases 2 and 4)". OEIS Foundation. https://oeis.org/A097856.
- ↑ Sloane, N. J. A., ed. "Sequence A000695 (Moser-de Bruijn sequence)". OEIS Foundation. https://oeis.org/A000695.
- ↑ Sloane, N. J. A., ed. "Sequence A006003". OEIS Foundation. https://oeis.org/A006003.
- ↑ 10.0 10.1 10.2 Sloane, N. J. A., ed. "Sequence A005917 (Rhombic dodecahedral numbers)". OEIS Foundation. https://oeis.org/A005917.
- ↑ Gould, H. W. (1978). "Euler's formula for [math]\displaystyle{ n }[/math]th differences of powers". The American Mathematical Monthly 85 (6): 450–467. doi:10.1080/00029890.1978.11994613.
- ↑ Jiang, Aiqin; Tyson, Trevor A.; Axe, Lisa (September 2005). "The structure of small Ta clusters". Journal of Physics: Condensed Matter 17 (39): 6111–6121. doi:10.1088/0953-8984/17/39/001. Bibcode: 2005JPCM...17.6111J.
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