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# 55

Please do not rely on any information it contains.

55 is an integer.

## Membership in core sequences

 Odd numbers ..., 49, 51, 53, 55, 57, 59, 61, ... A005843 Squarefree numbers ..., 59, 61, 62, 55, 66, 67, 69, ... A005117 Semiprimes ..., 46, 49, 51, 55, 57, 58, 62, ... A001358 Composite numbers ..., 51, 52, 54, 55, 56, 57, 58, ... A002808 Fibonacci numbers ..., 13, 21, 34, 55, 89, 144, 233, ... A000045 Triangular numbers ..., 28, 36, 45, 55, 66, 78, 91, ... A000217 Square pyramidal numbers ..., 5, 14, 30, 55, 91, 140, 204, ... A000330 ${\frac {\binom {3n}{n}}{2n+1}}$ 1, 1, 3, 12, 55, 273, 1428, ... A001764

Notice that 55 is the largest triangular number in the Fibonacci sequence, and in both sequences occurs at position 10. In Pascal's triangle, 55 occurs four times, two of those times being in the triangular numbers columns.

## Sequences pertaining to 55

 Multiples of 55 0, 55, 110, 165, 220, 275, 330, 385, 440, 495, 550, ... $3x+1$ sequence starting at 97 97, 292, 146, 73, 220, 110, 55, 166, 83, 250, 125, ... A008873

## Partitions of 55

There are 451276 partitions of 55. Of these, only the of the [FINISH WRITING]

## Roots and powers of 55

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\sqrt {55}}$ 7.41619848 A010508 55 2 3025 ${\sqrt[{3}]{55}}$ 3.80295246 A010626 55 3 166375 ${\sqrt[{4}]{55}}$ 2.72326981 A011048 55 4 9150625 ${\sqrt[{5}]{55}}$ 2.22880738 A011140 55 5 27680640625 ${\sqrt[{6}]{55}}$ 1.95011601 55 6 1522435234375 ${\sqrt[{7}]{55}}$ 1.77265100 55 7 83733937890625 ${\sqrt[{8}]{55}}$ 1.65023326 55 8 4605366583984375 ${\sqrt[{9}]{55}}$ 1.56089479 55 9 253295162119140625 ${\sqrt[{10}]{55}}$ 1.49291908 55 10 13931233916552734375

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## Factorization of some small integers in a quadratic integer ring adjoining the square roots of 55 and −55

The commutative quadratic integer ring with unity $\mathbb {Z} [{\sqrt {55}}]$ , with units of the form $\pm (89+12{\sqrt {55}})^{n}\,$ ($n\in \mathbb {Z}$ ), is not a unique factorization domain. Neither is ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-55}})}$ .

 $n$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-55}})}$ $\mathbb {Z} [{\sqrt {55}}]$ 2 Irreducible 3 4 2 2 5 Irreducible $(15-2{\sqrt {55}})(15+2{\sqrt {55}})$ 6 2 × 3 2 × 3 OR $(-1)(7-{\sqrt {55}})(7+{\sqrt {55}})$ 7 Irreducible 8 2 3 9 3 2 3 2 OR $(8-{\sqrt {55}})(8+{\sqrt {55}})$ 10 2 × 5 $2(15\pm 2{\sqrt {55}})$ 11 Irreducible $(-1)(22-3{\sqrt {55}})(22+3{\sqrt {55}})$ 12 2 2 × 3 13 Irreducible Irreducible despite positive Legendre symbol 14 2 × 7 OR $\left({\frac {1}{2}}-{\frac {\sqrt {-55}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {-55}}{2}}\right)$ 2 × 7 15 3 × 5 $3(15\pm 2{\sqrt {55}})$ 16 2 4 OR $\left({\frac {3}{2}}-{\frac {\sqrt {-55}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {-55}}{2}}\right)$ 2 4 17 Irreducible Irreducible despite positive Legendre symbol 18 2 × 3 2 19 Irreducible $(-1)(6-{\sqrt {55}})(6+{\sqrt {55}})$ 20 2 2 × 5 OR $\left({\frac {5}{2}}-{\frac {\sqrt {-55}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {-55}}{2}}\right)$ $2^{2}(15\pm 2{\sqrt {55}})$ ## Factorization of 55 in some quadratic integer rings

As was mentioned above, 55 is a squarefree semiprime in $\mathbb {Z}$ . But it has different factorizations in some quadratic integer rings.

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## Representation of 55 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 110111 2001 313 210 131 106 67 61 59 50 47 43 3D 3A 37 34 31 2H 2F

 $-1$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729