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# Hardy–Ramanujan number

1729 is the Hardy–Ramanujan number (taxi-cab number or taxicab number), the smallest [positive] integer that is the sum of 2 cubes in two different ways, viz.

${\displaystyle 1729\,=\,12^{3}+1^{3}\,=\,10^{3}+9^{3}.\,}$

## Other properties of 1729

Some properties of 1729:

## Roots and powers of 1729

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

Roots Value A-number Powers Value A-number
${\displaystyle {\sqrt {1729}}}$ 41.58124577 1729 2 2989441
${\displaystyle {\sqrt[{3}]{1729}}}$ 12.00231436 A215889 1729 3 5168743489
1729n, n ≥ 0.   A138130

## Sequences pertaining to 1729

 Multiples of 1729 0, 1729, 3458, 5187, 6916, 8645, 10374, 12103, 13832, 15561, ... A138129 1729-gonal numbers 1, 1729, 5184, 10366, 17275, 25911, 36274, 48364, 62181, 77725, ... A051871 ${\displaystyle 3x+1}$ sequence starting at 1729 1729, 5188, 2594, 1297, 3892, 1946, 973, 2920, 1460, 730, 365, ... A245671

## Values for number theoretic functions with 1729 as an argument

 ${\displaystyle \mu (1729)}$ –1 See Möbius function. ${\displaystyle M(1729)}$ –8 See Mertens function. ${\displaystyle \pi (1729)}$ 269 See prime counting function. ${\displaystyle \sigma _{0}(1729)}$ 8 See number of divisors function. ${\displaystyle \sigma _{1}(1729)}$ 2240 See sum of divisors function. ${\displaystyle \phi (1729)}$ 1296 See totient function. ${\displaystyle \Omega (1729)}$ 3 See number of prime factors (with multiplicity) function. ${\displaystyle \omega (1729)}$ 3 See number of distinct prime factors function. ${\displaystyle \lambda (1729)}$ 36 See Carmichael lambda function. ${\displaystyle \lambda (1729)}$ –1 See Liouville lambda function. ${\displaystyle \zeta (1729)}$ 1 + 3.30474152... × 10 –521 See Riemann zeta function. (Requires more than five hundred decimal places to distinguish from 1.) 1729! 1.86377... × 10 4849 See factorial. ${\displaystyle \Gamma (1729)}$ 1.0779473... × 10 4846 See Gamma function.

## Factorization of 1729 in some quadratic integer rings

INCOMPLETE. SOME CELLS IN THE TABLE BELOW ARE BLANK FOR NOW.

As was mentioned above, 1729 is a composite number in ${\displaystyle \mathbb {Z} }$. It is also composite in all quadratic integer rings, but its factorization differs, and it has multiple factorizations in some rings that are not unique factorization domains.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (-i)(2+3i)(3+2i)\times 7\times 19}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle 7\times 13(1-3{\sqrt {-2}})(1+3{\sqrt {-2}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle (3-{\sqrt {2}})(3+{\sqrt {2}})13\times 19}$ ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle (2-\omega )(2-\omega ^{2})(3-\omega )(3-\omega ^{2})(3-2\omega )(3-2\omega ^{2})}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ 7 × 13 × 19 ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle 7\times 13(5-\phi )(4+\phi )}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle (-1)({\sqrt {-7}})^{2}13\times 19}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (-1)({\sqrt {7}})^{2}13(18-7{\sqrt {7}})(18+7{\sqrt {7}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ ${\displaystyle 7\times 13(3-{\sqrt {-10}})(3+{\sqrt {-10}})}$OR ${\displaystyle (17-12{\sqrt {-10}})(17+12{\sqrt {-10}})}$OR ${\displaystyle (37-6{\sqrt {-10}})(37+6{\sqrt {-10}})}$Note that ${\displaystyle (27-10{\sqrt {-10}})(27+10{\sqrt {-10}})}$ is not a distinct factorization because ${\displaystyle 3+{\sqrt {-10}}}$ is a divisor of ${\displaystyle 27-10{\sqrt {-10}}}$. The same goes for ${\displaystyle (33-8{\sqrt {-10}})(33+8{\sqrt {-10}})}$. ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$

A side note: some Hilbert numbers have multiple factorizations, but 1729 is not one of them, being uniquely factorable into Hilbert numbers as 13 × 133.

 ${\displaystyle -1}$ 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