1105 (number)

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 ${\displaystyle {\tfrac {n(n^{2}+1)}{2}}}$ 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 (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]