# 1105 (number)

1105 (eleven hundred [and] five, or one thousand one hundred [and] five) is the natural number following 1104 and preceding 1106.

 ← 1104 1105 1106 →
Cardinalone thousand one hundred five
Ordinal1105th
(one thousand one hundred fifth)
Factorization5 × 13 × 17
Greek numeral,ΑΡΕ´
Roman numeralMCV
Binary100010100012
Ternary11112213
Octal21218
Duodecimal78112

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 ${\displaystyle {\tfrac {n(n^{2}+1)}{2}}}$ for ${\displaystyle n={}}$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, 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

1. ^ 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.
2. ^ 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. 13. Berlin: Springer. pp. 268–275. doi:10.1007/978-3-642-60408-9_21. MR 1425191.
3. ^
4. ^ Sloane, N. J. A. (ed.). "Sequence A002997 (Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. ^ 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. 9. Springer-Verlag, New York. p. 136. doi:10.1007/978-0-387-21850-2. ISBN 0-387-95332-9. MR 1866957.
6. ^ Carmichael, R. D. (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16: 232–238. doi:10.1090/S0002-9904-1910-01892-9. JFM 41.0226.04.
7. ^ 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.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000695 (Moser-de Bruijn sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. ^ Sloane, N. J. A. (ed.). "Sequence A006003". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. ^ a b c Sloane, N. J. A. (ed.). "Sequence A005917 (Rhombic dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
11. ^ Gould, H. W. (1978). "Euler's formula for ${\displaystyle n}$ th differences of powers". The American Mathematical Monthly. 85 (6): 450–467. doi:10.1080/00029890.1978.11994613. JSTOR 2320064. MR 0480057.
12. ^ 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.