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 $(6m+1)\cdot (12m+1)\cdot (18m+1),m\ge 1.$)
• First extended Chernick's Carmichael number.
• First absolute Euler pseudoprime (A033181).
• Third Zeisel number (A051015).
• 1729 is some figurate number, e.g. the 19^th 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
			1729 n, 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.
$\varphi (1729)$	1296	See totient function.
$\mathrm{\Omega }(1729)$	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.
$\mathrm{\Gamma }(1729)$	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 $\mathbb{\mathbb{Z}}$. 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.

$\mathbb{\mathbb{Z}}[i]$	$(-i)(2+3i)(3+2i)\times 7\times 19$
$\mathbb{\mathbb{Z}}[\sqrt{-2}]$	$7\times 13(1-3\sqrt{-2})(1+3\sqrt{-2})$	$\mathbb{\mathbb{Z}}[\sqrt{2}]$	$(3-\sqrt{2})(3+\sqrt{2})13\times 19$
$\mathbb{\mathbb{Z}}[\omega ]$	$(2-\omega )(2-{\omega }^2)(3-\omega )(3-{\omega }^2)(3-2\omega )(3-2{\omega }^2)$	$\mathbb{\mathbb{Z}}[\sqrt{3}]$
$\mathbb{\mathbb{Z}}[\sqrt{-5}]$	7 × 13 × 19	$\mathbb{\mathbb{Z}}[\varphi ]$	$7\times 13(5-\varphi )(4+\varphi )$
$\mathbb{\mathbb{Z}}[\sqrt{-6}]$		$\mathbb{\mathbb{Z}}[\sqrt{6}]$
${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-7})}$	$(-1)(\sqrt{-7}{)}^{2}13\times 19$	$\mathbb{\mathbb{Z}}[\sqrt{7}]$	$(-1)(\sqrt{7}{)}^{2}13(18-7\sqrt{7})(18+7\sqrt{7})$
$\mathbb{\mathbb{Z}}[\sqrt{-10}]$	$7\times 13(3-\sqrt{-10})(3+\sqrt{-10})$ 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 $(33-8\sqrt{-10})(33+8\sqrt{-10})$.	$\mathbb{\mathbb{Z}}[\sqrt{10}]$

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.

See also

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

Some integers

							$-1$
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