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# 73

Please do not rely on any information it contains.

73 is the 21st prime number, and in base 10 it happens that the 12th prime number is 37.

## Membership in core sequences

 Odd numbers ..., 67, 69, 71, 73, 75, 77, 79, ... A005843 Prime numbers ..., 61, 67, 71, 73, 79, 83, 89, ... A000040 Lucky numbers ..., 63, 67, 69, 73, 75, 79, 87, ... A000959 Squarefree numbers ..., 69, 70, 71, 73, 74, 77, 78, ... A005117 Numbers that are the sum of two squares ..., 65, 68, 72, 73, 74, 80, 81, ... A001481 Loeschian numbers ..., 63, 64, 67, 73, 75, 76, 79, ... A003136

## Sequences pertaining to 73

 Multiples of 73 73, 146, 219, 292, 365, 438, 511, 584, 657, 730, 803, 876, ... $3x+1$ sequence beginning at 73 73, 220, 110, 55, 166, 84, 42, 21, 64, 32, 16, 8, 4, 2, 1, ... $5x+1$ sequence beginning at 11 ..., 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, ...

## Partitions of 73

There are 6185689 partitions of 77. Of these, the of the [FINISH WRITING]

## Roots and powers of 73

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

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REMARKS

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## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −73, 73

The commutative quadratic integer ring with unity ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {73}})}$ , with units of the form $\pm (1068+125{\sqrt {73}})^{n}\,$ ($n\in \mathbb {Z}$ ), is not only a unique factorization domain, it is also a norm-Euclidean domain, and in fact no quadratic integer ring with higher discriminant is norm-Euclidean (but there are higher with unique factorization).

 $n$ ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {73}})}$ 2 $\left({\frac {9}{2}}-{\frac {\sqrt {73}}{2}}\right)\left({\frac {9}{2}}+{\frac {\sqrt {73}}{2}}\right)$ 3 $(-1)(17-2{\sqrt {73}})(17+2{\sqrt {73}})$ 4 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)^{2}$ 5 Prime 6 $(-1)\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)(17\pm 2{\sqrt {73}})$ 7 Prime 8 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)^{3}$ 9 $(17\pm 2{\sqrt {73}})^{2}$ 10 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)5$ 11 Prime 12 $(-1)\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)^{2}(17\pm 2{\sqrt {73}})$ 13 Prime 14 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)7$ 15 $(17\pm 2{\sqrt {73}})5$ 16 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)^{4}$ 17 Prime 18 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)(17\pm 2{\sqrt {73}})^{2}$ 19 $(26-3{\sqrt {73}})(26+3{\sqrt {73}})$ 20 $\left({\frac {9}{2}}\pm {\frac {\sqrt {73}}{2}}\right)^{2}5$ Note that $\left({\frac {7}{2}}\pm {\frac {\sqrt {73}}{2}}\right)$ does not constitute a distinct factorization of 6 since $\left({\frac {9}{2}}-{\frac {\sqrt {73}}{2}}\right)(17+2{\sqrt {73}})={\frac {7}{2}}+{\frac {\sqrt {73}}{2}}$ .

$\mathbb {Z} [{\sqrt {-73}}]$ is not a unique factorization domain. But the window of 1 through 20 does not provide as interesting a window for the of the [FINISH WRITING]

PLACEHOLDER

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## Representation of 73 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 1001001 2201 1021 243 201 133 111 81 73 67 61 58 53 4D 49 45 41 3G 3D

 $-1$ 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