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:

$1105 = 33^2 + 4^2 = 32^2 + 9^2 = 31^2 + 12^2 = 24^2 + 23^2$