1105

Number
$1105$ (one thousand, one hundred and five) is:


 * $5 \times 13 \times 17$


 * The $2$nd Carmichael number after $561$:
 * $\forall a \in \Z: a \perp 1105: a^{1104} \equiv 1 \pmod {1105}$


 * The $4$th Poulet number after $341$, $561$, $645$:
 * $2^{1105} \equiv 2 \pmod {1105}$: $1105 = 5 \times 13 \times 17$


 * The $7$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$:
 * $3^{1105} \equiv 3 \pmod {1105}$


 * The $10$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$, $561$, $645$, $703$:
 * $4^{1105} \equiv 4 \pmod {1105}$


 * The magic constant of a magic square of order $13$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$, $505$, $671$, $870$:
 * $1105 = \ds \dfrac 1 {13} \sum_{k \mathop = 1}^{13^2} k = \dfrac {13 \paren {13^2 + 1} } 2$


 * The product of the first $3$ primes of the form $4 n + 1$:
 * $1105 = \paren {4 + 1} \paren {12 + 1} \paren {16 + 1}$


 * Can be expressed as the sum of two squares in more ways than any smaller integer:
 * $1105 = 33^2 + 4^2 = 32^2 + 9^2 = 31^2 + 12^2 = 24^2 + 23^2$

Also see

 * 1105 as Sum of Two Squares