The world of mathematics is renowned with number of interesting and fascinating numbers.Now Ramanujan Number also makes such place in the list. Ramanujan Numbers( preciously termed as Hardy-Ramanujan Numbers) are those numbers which are the smallest positive integers which can be represented or expressed as a sum of 2 positive integers in n ways.

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,…….

——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 this 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)

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.