An interesting property

Number Theory Level pending

What is the least positive number that is the sum of two numbers to the third power - in two different ways?


The answer is 1729.

Victoria Willis by Victoria Willis , 56, USA

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1 solution

1729 = 1 2 3 + 1 3 = 1 0 3 + 9 3 1729=12^3+1^3=10^3+9^3

However, it is NOT the only number with such a property. There are infinitely many such numbers. Few examples are

2 3 + 1 6 3 = 9 3 + 1 5 3 = 4104 2^3 + 16^3 = 9^3 + 15^3 =4104

2 3 + 2 4 3 = 1 8 3 + 2 0 3 = 13832 2^3 + 24^3 = 18^3 + 20^3 =13832

1 0 3 + 2 7 3 = 1 9 3 + 2 4 3 = 20683 10^3 + 27^3 = 19^3 + 24^3 =20683

4 3 + 3 2 3 = 1 8 3 + 3 0 3 = 32832 4^3 + 32^3 = 18^3 + 30^3 =32832

2 3 + 3 4 3 = 1 5 3 + 3 3 3 = 39312 2^3 + 34^3 = 15^3 + 33^3 =39312

9 3 + 3 4 3 = 1 6 3 + 3 3 3 = 40033 9^3 + 34^3 = 16^3 + 33^3 =40033

3 3 + 3 6 3 = 2 7 3 + 3 0 3 = 46683 3^3 + 36^3 = 27^3 + 30^3 =46683

1 7 3 + 3 9 3 = 2 6 3 + 3 6 3 = 64232 17^3 + 39^3 = 26^3 + 36^3 =64232

1 2 3 + 4 0 3 = 3 1 3 + 3 3 3 = 65728 12^3 + 40^3 = 31^3 + 33^3 =65728

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You're right, it is the least. My math teacher was wrong!

Victoria Willis - 6 years, 4 months ago

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