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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?

Why is 1729 a Special 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
what are the first three ramanujan numbers

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)

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