1105 (number)

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