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 thesum of two squares theorem) to its factorization5 × 13 × 17 as the product of the three smallestprime numbers that are congruent to 1 modulo 4.[2][3] It is also the smallest member of a cluster of threesemiprimes (1105, 1106, 1107) with eightdivisors,[4] and the second-smallestCarmichael number, after561,[5][6] one of the first four Carmichael numbers identified byR. D. Carmichael in his 1910 paper introducing this concept.[6][7]
As a number of the form for13, 1105 is themagic constant for13 × 13magic squares,[10] and as a difference of two consecutive fourth powers(1105 = 74 − 64)[11][12] it is a rhombic dodecahedral number (a type offigurate number), and amagic number forbody-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 (here7 + 6 = 13).[11]
