1105
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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
- Previous ... Next: Carmichael Number
- Previous ... Next: Poulet Number
- Previous ... Next: Fermat Pseudoprime to Base 4
- Previous ... Next: Magic Constant of Magic Square
- Previous ... Next: Fermat Pseudoprime to Base 3
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1105$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1105$