What are the first three Ramanujan numbers?

The world of mathematics is renowned for a number of interesting and fascinating numbers. Now Ramanujan Number also makes such a place in the list. Ramanujan Numbers (preciously termed as Hardy-Ramanujan Numbers) are those numbers that are the smallest positive integers that can be represented or expressed as a sum of 2 positive integers in n ways. Before discussing what are the first three Ramanujan numbers and how I found them in an amazing way, let's understand why is 1729 a special number.

Why is 1729 a special number and why is it called Hardy-Ramanujan number?

Now this Ramanujan’s n-way solution is a way in which a positive integer that can be expressed as a sum of 2 cubes in n different ways.

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1. Ramanujan’s 1-way solution
2. Ramanujan’s 2-way solution
3. Ramanujan’s 3-way solution
4. Ramanujan’s 4-way solution
5. Ramanujan’s 5-way solution
6. Ramanujan’s 6-way solution

Now let’s discuss the above ways in a mathematical way.

Ramanujan’s 1-way solution

Integers that are expressed as the sum of 2 cubes (in at least one way). Some of these numbers include :

{2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, ...}

——Verifying——

2=\(1^3\)+\(1^3\)

9=\(2^3\)+\(1^3\)

16=\(2^3\)+\(2^3\)

Here all these numbers can be expressed as a sum of 2 cubes in a single way and so all these numbers from the above set can be expressed in this way.

Ramanujan’s 2-way solution

Integers that can be expressed as sum of 2 cubes in more than 1 way(at least in 2 ways) . Some of these numbers includes

{1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, ...}

——Verifying——-

1729=\(12^3\)+\(1^3\)=\(10^3\)+\(9^3\)

4104=\(16^3\)+\(2^3\)=\(15^3\)+\(9^3\)

13832=\(24^3\)+\(2^3\)=\(20^3\)+\(18^3\)

20683=\(27^3\)+\(10^3\)=\(24^3\)+\(19^3\)

Here all these numbers can be expressed as a sum of 2 cubes in 2 different ways . All of these numbers form the set can be expressed in these ways.

Ramanujan’s 3-way solution

Integers that can be expressed as sum of 2 cubes in at least 3 ways. Some of these numbers includes

{87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, ...}

——Verifying——-

87539319=\(167^3\)+\(436^3\)=\(228^3\)+\(423^3\)=\(255^3\)+\(414^3\)

Here all the numbers from the above set can be expressed as a sum of 2 cubes in 3 ways.

Ramanujan’s 4-way solution

Integers that can be expressed as sum of 2 cubes in at least 4 ways. Some of these numbers includes

{6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, ...}

——Verifying——

6963472309248=\(2421^3\)+\(19083^3\)=\(5436^3\)+\(18948^3\)=\(10200^3\)+\(18072^3\)=\(13322^3\)+\(16630^3\)

Here all the numbers from the above set can be expressed as a sum of 2 cubes in 4 ways.

Ramanujan’s 5-way solutions

Integers that can be expressed as sum of 2 cubes in at least 5 ways. Some of these numbers are

{48988659276962496, 391909274215699968, 490593422681271000, 1322693800477987392, 3135274193725599744, 3924747381450168000, 6123582409620312000, 6355491080314102272, ...}

—-Verifying——

48988659276962496=\(38787^3\)+\(365757^3\)=\(107839^3\)+\(363753^3\)=\(205292^3\)+\(342952^3\)=\(221424^3\)+\(336588^3\)=\(231518^3\)+\(331954^3\)

Here this numbers can be expressed as a sum of 2 cubes in 5 different ways . All of these numbers of the above set can be expressed in these ways.

Ramanujan’s 6-way solutions

Integers that can be expressed as a sum of 2 cubes in at least 6 ways. Example

24153319581254312065344,…….

So these are all the numbers, now for the first three Ramanujan’s numbers, one can check for the type first, then can see the first 3 numbers.

——Verifying——

24153319581254312065344=\(582162^3\)+\(28906206^3\)=\(3064173^3\)+\(28894803^3\)=\(8519281^3\)+\(28657487^3\)=\(16218068^3\)+\(27093208^3\)=\(17492496^3\)+\(26590452^3\)=\(18289922^3\)+\(26224366^3\)

Here these numbers can be expressed as a sum of 2 cubes in 6 different ways . All of these numbers of the above set can be expressed in these ways.

All this numbers were also known as taxicab numbers and also denoted by Ta(n)

Some Useful Links:

• Triangular Number Sequence - Explanation with examples
• Inversion of a line - Problem video
• Pigeon Hole Principle