1105 (eleven hundred [and] five, or one thousand one hundred [and] five) is the natural number following 1104 and preceding 1106.
| ||||
|---|---|---|---|---|
| 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 | |||
| Senary | 50416 | |||
| Octal | 21218 | |||
| Duodecimal | 78112 | |||
| Hexadecimal | 45116 | |||
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 smallest member of a cluster of three semiprimes (1105, 1106, 1107) with eight divisors,[4] and the second-smallest Carmichael number, after 561,[5][6] one of the first four Carmichael numbers identified by R. D. Carmichael in his 1910 paper introducing this concept.[6][7]
Its binary representation 10001010001 and its base-4 representation 101101 are both palindromes,[8] 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.[9]
As a number of the form for 13, 1105 is the magic constant for 13 × 13 magic squares,[10] and as a difference of two consecutive fourth powers (1105 = 74 − 64)[11][12] it is a rhombic dodecahedral number (a type of figurate number), and a magic number for body-centered cubic crystals.[11][13] 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).[11]
References
edit- ^ Sloane, N. J. A. (ed.). "Sequence A016032 (Least positive integer that is the sum of two squares of positive integers in exactly n ways)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Tenenbaum, Gérald (1997). "1105: first steps in a mysterious quest". In Graham, Ronald L.; Nešetřil, Jaroslav (eds.). The mathematics of Paul Erdős, I. Algorithms and Combinatorics. Vol. 13. Berlin: Springer. pp. 268–275. doi:10.1007/978-3-642-60408-9_21. ISBN 978-3-642-64394-1. MR 1425191.
- ^ Sloane, N. J. A. (ed.). "Sequence A006278 (product of the first n primes congruent to 1 (mod 4))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A005238 (Numbers k such that k, k+1 and k+2 have the same number of divisors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A002997 (Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b 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. Vol. 9. Springer-Verlag, New York. p. 136. doi:10.1007/978-0-387-21850-2. ISBN 0-387-95332-9. MR 1866957.
- ^ 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. JFM 41.0226.04.
- ^ Sloane, N. J. A. (ed.). "Sequence A097856 (Numbers that are palindromic in bases 2 and 4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000695 (Moser-de Bruijn sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A006003". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A005917 (Rhombic dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Gould, H. W. (1978). "Euler's formula for th differences of powers". The American Mathematical Monthly. 85 (6): 450–467. doi:10.1080/00029890.1978.11994613. JSTOR 2320064. MR 0480057.
- ^ Jiang, Aiqin; Tyson, Trevor A.; Axe, Lisa (September 2005). "The structure of small Ta clusters". Journal of Physics: Condensed Matter. 17 (39): 6111–6121. Bibcode:2005JPCM...17.6111J. doi:10.1088/0953-8984/17/39/001. S2CID 41954369.