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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?

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)

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