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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.

$1729\,=\,12^{3}+1^{3}\,=\,10^{3}+9^{3}.\,$ ## Other properties of 1729

Some properties of 1729:

• Third Carmichael number (absolute Fermat pseudoprime) (A002997).
• First Chernick's Carmichael number (A033502). (1729 = 7 × 13 × 19 is the smallest Carmichael number of the form $(6m+1)\cdot (12m+1)\cdot (18m+1),\,m\,\geq \,1.\,$ )
• First extended Chernick's Carmichael number.
• First absolute Euler pseudoprime (A033181).
• Third Zeisel number (A051015).
• 1729 is some figurate number, e.g. the 19th dodecagonal number (A051624) since 1729 = 19 × 91. It happens that the first and last digits of 1729 give 19, and the middle two digits 72 = 6 × 12 = (7 − 1) × (13 − 1), 7 × 13 = 91 being the reversal of 19.
• Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 1 (trivial case), 81 and 1458) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number: 1 + 7 + 2 + 9 = 19; 19 × 91 = 1729.
• ${\frac {1729+9271}{11}}\,=\,10^{3}.$ ## 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
${\sqrt {1729}}$ 41.58124577 1729 2 2989441
${\sqrt[{3}]{1729}}$ 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 $3x+1$ 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

 $\mu (1729)$ –1 See Möbius function. $M(1729)$ –8 See Mertens function. $\pi (1729)$ 269 See prime counting function. $\sigma _{0}(1729)$ 8 See number of divisors function. $\sigma _{1}(1729)$ 2240 See sum of divisors function. $\phi (1729)$ 1296 See totient function. $\Omega (1729)$ 3 See number of prime factors (with multiplicity) function. $\omega (1729)$ 3 See number of distinct prime factors function. $\lambda (1729)$ 36 See Carmichael lambda function. $\lambda (1729)$ –1 See Liouville lambda function. $\zeta (1729)$ 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. $\Gamma (1729)$ 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 $\mathbb {Z}$ . 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.

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

• A011541 Taxi-cab (taxicab) or Hardy–Ramanujan numbers: the smallest number that is the sum of 2 cubes in $n\,$ ways (an infinite sequence).
 $-1$ 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