1729 (number)
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Cardinal | one thousand seven hundred twenty-nine | |||
Ordinal | 1729th (one thousand seven hundred twenty-ninth) |
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Factorization | 7 × 13 × 19 | |||
Divisors | 1, 7, 13, 19, 91, 133, 247, 1729 | |||
Greek numeral | ,ΑΨΚΘ´ | |||
Roman numeral | MDCCXXIX | |||
Binary | 11011000001_{2} | |||
Ternary | 2101001_{3} | |||
Quaternary | 123001_{4} | |||
Quinary | 23404_{5} | |||
Senary | 12001_{6} | |||
Octal | 3301_{8} | |||
Duodecimal | 1001_{12} | |||
Hexadecimal | 6C1_{16} | |||
Vigesimal | 469_{20} | |||
Base 36 | 1C1_{36} |
1729 is the natural number following 1728 and preceding 1730. It is known as the Hardy-Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:^{[1]}^{[2]}^{[3]}^{[4]}
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
The two different ways are:
- 1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729):
- 91 = 6^{3} + (−5)^{3} = 4^{3} + 3^{3}
Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways^{[5]} have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.
The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in the OEIS) defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z^{3} which are also expressible as the sum of two other cubes.
Other properties
1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.
1729 is a Zeisel number.^{[6]} It is a centered cube number,^{[7]} as well as a dodecagonal number,^{[8]} a 24-gonal^{[9]} and 84-gonal number.
Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729 (Guy 2004).
Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal (1729 = 3301_{8}, 3 + 3 + 0 + 1 = 7) and hexadecimal (1729 = 6C1_{16}, 6 + C + 1 = 19_{10}), but not in binary and duodecimal.
In base 12, 1729 is written as 1001, so its reciprocal has only period 6 in that base.
1729 is the lowest number which can be represented by a Loeschian quadratic form a² + ab + b² in four different ways with a and b positive integers. The integer pairs (a,b) are (25,23), (32,15), (37,8) and (40,3).^{[10]}
1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first consecutive occurrence of all ten digits without repetition in the decimal representation of the transcendental number e.^{[11]}
Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) 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
It suffices only to check sums congruent to 0 or 1 (mod 9) up to 19.
See also
- A Disappearing Number, a 2007 play about Ramanujan in England during World War I.
- Berry paradox
- Interesting number paradox
- Taxicab number
- 4104, the second positive integer which can be expressed as the sum of two positive cubes in two different ways.
References
- Gardner, Martin (1973), Mathematical Puzzles and Diversions (Paperback ed.), Pelican / Penguin Books, ISBN 0-14-020713-9
- Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, Vol. 1 (3rd ed.), Springer, ISBN 0-387-20860-7 - D1 mentions the Hardy–Ramanujan number.
Notes
- ^ Quotations by Hardy Archived 2012-07-16 at the Wayback Machine.
- ^ Singh, Simon (15 October 2013). "Why is the number 1,729 hidden in Futurama episodes?". BBC News Online. Retrieved 15 October 2013.
- ^ Hardy, G H (1940). Ramanujan. New York: Cambridge University Press (original). p. 12.
- ^ Hardy, G. H. (1921), "Srinivasa Ramanujan", Proc. London Math. Soc., s2-19 (1): xl–lviii, doi:10.1112/plms/s2-19.1.1-u The anecdote about 1729 occurs on pages lvii and lviii
- ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 13. ISBN 978-1-84800-000-1.
- ^ Sloane, N.J.A. (ed.). "Sequence A051015 (Zeisel numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ Sloane, N.J.A. (ed.). "Sequence A005898 (Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ Sloane, N.J.A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ Sloane, N.J.A. (ed.). "Sequence A051876 (24-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02.
- ^ David Mitchell (25 February 2017). "Tessellating the Hardy-Ramanujan Taxicab Number, 1729, Bedrock of Integer Sequence A198775". Retrieved 19 July 2018.
- ^ The Dullness of 1729
External links
- MathWorld: Hardy–Ramanujan Number
- BBC: A Further Five Numbers
- Grime, James; Bowley, Roger. "1729: Taxi Cab Number or Hardy-Ramanujan Number". Numberphile. Brady Haran.
- Why does the number 1729 show up in so many Futurama episodes?, io9.com