Speciality Of The Number 1729...?

[Anbe_shivam1]

Evariki ina telusa...?

[Anbe_shivam1]

Evariki telida... aa number enduku special

guess cheyyandi

[Jatka Bandi]

[b]1729[/b] is the [url="http://en.wikipedia.org/wiki/Natural_number"]natural number[/url] following [url="http://en.wikipedia.org/wiki/1728_%28number%29"]1728[/url] and preceding 1730. [b]1729[/b] is known as the [b]Hardy–Ramanujan number[/b] after a famous anecdote of the British mathematician [url="http://en.wikipedia.org/wiki/G._H._Hardy"]G. H. Hardy[/url] regarding a hospital visit to the Indian mathematician [url="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan"]Srinivasa Ramanujan[/url]. In Hardy's words:[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-0"][1][/url][/sup] “ I remember once going to see him when he was ill at [url="http://en.wikipedia.org/wiki/Putney"]Putney[/url]. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a [url="http://en.wikipedia.org/wiki/Interesting_number_paradox"]dull one[/url], and that I hoped it was not an unfavorable [url="http://en.wikipedia.org/wiki/Omen"]omen[/url]. "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 these: 1729 = 1[sup]3[/sup] + 12[sup]3[/sup] = 9[sup]3[/sup] + 10[sup]3[/sup]
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a [url="http://en.wikipedia.org/wiki/Negative_number"]negative[/url] [url="http://en.wikipedia.org/wiki/Integer"]integer[/url]) gives the smallest solution as [url="http://en.wikipedia.org/wiki/91_%28number%29"]91[/url] (which is a divisor of 1729): 91 = 6[sup]3[/sup] + (−5)[sup]3[/sup] = 4[sup]3[/sup] + 3[sup]3[/sup]
Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".
Numbers that are the smallest number that can be expressed as the sum of two cubes in [i]n[/i] distinct ways[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-1"][2][/url][/sup] have been dubbed "[url="http://en.wikipedia.org/wiki/Taxicab_number"]taxicab numbers[/url]". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by [url="http://en.wikipedia.org/wiki/Fr%C3%A9nicle_de_Bessy"]Frénicle de Bessy[/url] in 1657.
The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence [url="http://oeis.org/A050794"]A050794[/url] in [url="http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences"]OEIS[/url]) defined as numbers of the form 1 + [i]z[/i][sup]3[/sup] which are also expressible as the sum of two other cubes.
1729 is also the third [url="http://en.wikipedia.org/wiki/Carmichael_number"]Carmichael number[/url] and the first absolute [url="http://en.wikipedia.org/wiki/Euler_pseudoprime"]Euler pseudoprime[/url]. It is also a [url="http://en.wikipedia.org/wiki/Sphenic_number"]sphenic number[/url].
1729 is a [url="http://en.wikipedia.org/wiki/Zeisel_number"]Zeisel number[/url]. It is a [url="http://en.wikipedia.org/wiki/Centered_cube_number"]centered cube number[/url], as well as a [url="http://en.wikipedia.org/wiki/Dodecagonal_number"]dodecagonal number[/url], a 24-[url="http://en.wikipedia.org/wiki/Polygonal_number"]gonal[/url] and 84-gonal number.
Investigating pairs of distinct integer-valued [url="http://en.wikipedia.org/wiki/Quadratic_form"]quadratic forms[/url] 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 [url="http://en.wikipedia.org/wiki/Discriminant"]discriminant[/url] 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 [url="http://en.wikipedia.org/wiki/Harshad_number"]Harshad number[/url]. It also has this property in [url="http://en.wikipedia.org/wiki/Octal"]octal[/url] (1729 = 3301[sub]8[/sub], 3 + 3 + 0 + 1 = 7) and [url="http://en.wikipedia.org/wiki/Hexadecimal"]hexadecimal[/url] (1729 = 6C1[sub]16[/sub], 6 + C + 1 = 19[sub]10[/sub]), but not in [url="http://en.wikipedia.org/wiki/Binary_numeral_system"]binary[/url].
1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the [url="http://en.wikipedia.org/wiki/Transcendental_number"]transcendental number[/url] [url="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29"][i]e[/i][/url].[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-2"][3][/url][/sup]
[url="http://en.wikipedia.org/wiki/Masahiko_Fujiwara"]Masahiko Fujiwara[/url] showed that 1729 is one of four positive integers (with the others being [url="http://en.wikipedia.org/wiki/81_%28number%29"]81[/url], [url="http://en.wikipedia.org/wiki/1458_%28number%29"]1458[/url], and the trivial case [url="http://en.wikipedia.org/wiki/1_%28number%29"]1[/url]) 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.

More[i] [i]Info: [/i][/i]http://en.wikipedia.org/wiki/1729_%28number%29

[Anbe_shivam1]

[quote name='bantrothu bullabbai' timestamp='1341703577' post='1302096531']
[b]1729[/b] is the [url="http://en.wikipedia.org/wiki/Natural_number"]natural number[/url] following [url="http://en.wikipedia.org/wiki/1728_%28number%29"]1728[/url] and preceding 1730. [b]1729[/b] is known as the [b]Hardy–Ramanujan number[/b] after a famous anecdote of the British mathematician [url="http://en.wikipedia.org/wiki/G._H._Hardy"]G. H. Hardy[/url] regarding a hospital visit to the Indian mathematician [url="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan"]Srinivasa Ramanujan[/url]. In Hardy's words:[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-0"][1][/url][/sup] “ I remember once going to see him when he was ill at [url="http://en.wikipedia.org/wiki/Putney"]Putney[/url]. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a [url="http://en.wikipedia.org/wiki/Interesting_number_paradox"]dull one[/url], and that I hoped it was not an unfavorable [url="http://en.wikipedia.org/wiki/Omen"]omen[/url]. "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 these: 1729 = 1[sup]3[/sup] + 12[sup]3[/sup] = 9[sup]3[/sup] + 10[sup]3[/sup]
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a [url="http://en.wikipedia.org/wiki/Negative_number"]negative[/url] [url="http://en.wikipedia.org/wiki/Integer"]integer[/url]) gives the smallest solution as [url="http://en.wikipedia.org/wiki/91_%28number%29"]91[/url] (which is a divisor of 1729): 91 = 6[sup]3[/sup] + (−5)[sup]3[/sup] = 4[sup]3[/sup] + 3[sup]3[/sup]
Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".
Numbers that are the smallest number that can be expressed as the sum of two cubes in [i]n[/i] distinct ways[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-1"][2][/url][/sup] have been dubbed "[url="http://en.wikipedia.org/wiki/Taxicab_number"]taxicab numbers[/url]". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by [url="http://en.wikipedia.org/wiki/Fr%C3%A9nicle_de_Bessy"]Frénicle de Bessy[/url] in 1657.
The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence [url="http://oeis.org/A050794"]A050794[/url] in [url="http://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences"]OEIS[/url]) defined as numbers of the form 1 + [i]z[/i][sup]3[/sup] which are also expressible as the sum of two other cubes.
1729 is also the third [url="http://en.wikipedia.org/wiki/Carmichael_number"]Carmichael number[/url] and the first absolute [url="http://en.wikipedia.org/wiki/Euler_pseudoprime"]Euler pseudoprime[/url]. It is also a [url="http://en.wikipedia.org/wiki/Sphenic_number"]sphenic number[/url].
1729 is a [url="http://en.wikipedia.org/wiki/Zeisel_number"]Zeisel number[/url]. It is a [url="http://en.wikipedia.org/wiki/Centered_cube_number"]centered cube number[/url], as well as a [url="http://en.wikipedia.org/wiki/Dodecagonal_number"]dodecagonal number[/url], a 24-[url="http://en.wikipedia.org/wiki/Polygonal_number"]gonal[/url] and 84-gonal number.
Investigating pairs of distinct integer-valued [url="http://en.wikipedia.org/wiki/Quadratic_form"]quadratic forms[/url] 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 [url="http://en.wikipedia.org/wiki/Discriminant"]discriminant[/url] 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 [url="http://en.wikipedia.org/wiki/Harshad_number"]Harshad number[/url]. It also has this property in [url="http://en.wikipedia.org/wiki/Octal"]octal[/url] (1729 = 3301[sub]8[/sub], 3 + 3 + 0 + 1 = 7) and [url="http://en.wikipedia.org/wiki/Hexadecimal"]hexadecimal[/url] (1729 = 6C1[sub]16[/sub], 6 + C + 1 = 19[sub]10[/sub]), but not in [url="http://en.wikipedia.org/wiki/Binary_numeral_system"]binary[/url].
1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the [url="http://en.wikipedia.org/wiki/Transcendental_number"]transcendental number[/url] [url="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29"][i]e[/i][/url].[sup][url="http://en.wikipedia.org/wiki/1729_%28number%29#cite_note-2"][3][/url][/sup]
[url="http://en.wikipedia.org/wiki/Masahiko_Fujiwara"]Masahiko Fujiwara[/url] showed that 1729 is one of four positive integers (with the others being [url="http://en.wikipedia.org/wiki/81_%28number%29"]81[/url], [url="http://en.wikipedia.org/wiki/1458_%28number%29"]1458[/url], and the trivial case [url="http://en.wikipedia.org/wiki/1_%28number%29"]1[/url]) 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.

More[i] [i]Info: [/i][/i][url="http://en.wikipedia.org/wiki/1729_%28number%29"]http://en.wikipedia....i/1729_(number)[/url]
[/quote]

GP mama...

Srinivasa ramanujam

[bokkale]

Only number which can be expressed as sum of cubes of two successive numbers less than 10...