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Number Theory Level 2

Let n n be a positive integer such that

a 3 + b 3 = n a^{3}+b^{3}=n

c 3 + d 3 = n c^{3}+d^{3}=n

Where a a , b b , c c and d d are distinct, positive integers. What is the smallest possible value of n n ?


The answer is 1729.

Finn Hulse by Finn Hulse , 21, USA

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5 solutions

Venture Hi
Mar 19, 2014

Ramanujan's taxi number?

10 Helpful 0 Interesting 0 Brilliant 0 Confused

I have heard this in the autobiography of Ramanuja the great mathematician

Saurav Sah - 7 years, 2 months ago

Hardy-Ramanujan Number

Harikrishna Menon - 7 years, 1 month ago
Michael Mendrin
Mar 19, 2014

It has a name, the "Hardy-Ramanujan number". Lots of history and literature with this one.

6 Helpful 0 Interesting 0 Brilliant 0 Confused

I didn't try to solve this, I just recognized the problem and remembered the number.

Michael Mendrin - 7 years, 2 months ago

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Me too, I remembered that I solved this kind of problem last week. :)

Ahmad Naufal Hakim - 7 years, 2 months ago

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Could you post a solution or tell me where I can find one, please? I can't find the reasoning anywhere...

Jordi Pomada - 5 years, 10 months ago
Shreyas Shastry
Mar 19, 2014

1729 is the only number that can be represented as a sum of two cube numbers

10^3+9^3=1729

12^3+1^3=1729

so the answer is 1729

2 Helpful 1 Interesting 0 Brilliant 0 Confused

By the way, there are infinitely many numbers that can be represented as a sum of 2 cubes. It's just that 1729 is the smallest.

King Zhang Zizhong - 7 years, 2 months ago

you can easily write a program to find out as many solutions as possible.

Nikhil Supekar - 7 years, 2 months ago

Not the only one, the smallest one.

Yuxuan Seah - 7 years, 2 months ago

1729 is also special because it is the 3rd Carmichael number

Nanayaranaraknas Vahdam - 7 years, 2 months ago

NO, it is not the only number, but it is the smallest of this catgory

Kushagra Sahni - 7 years, 2 months ago

Guys I heard about it in a numberphile video but how do you solve it?

Adrian Neacșu - 7 years, 1 month ago

its ramanujan number and this speciality about the number was pointed out by the greatest giant of mathematics on the bed while talking with hardy when he was ill .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Sharky Kesa
Mar 19, 2014

Taxicab or Hardy-Ramanujan number. Click here to see my problem on it.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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