understanding Ramanujan number

It is  the smallest number that is  expressed by the sum of two cubes in two different ways. 1729 is sometimes called the “Hardy-Ramanujan number”. The story is well known to all. But now I want describe how they are formed.

The number should express in sum of two cubes in two different ways.

Now here we will see cubes of the numbers upto 16.

number	Cube of number
13	1
23	8
33	27
43	64
53	125
63	216
73	343
83	512
93	729
103	1000
113	1331
123	1728
133	2197
143	2744
153	3375
163	4096

Ramanujan number is expressed in sum of two cubes in two different ways.

It is expressed in sum of two cubes , we will take sum of two cubes in above 16 cubes.

If we take sum of two cubes from 16 numbers. We will get 16c2 combinations.

16c2=130 combinations are available.

We will observe all combinations in below table.

1+8=9	64+125=189	512+729=1241
1+27=28	64+216=280	512+1000=1512
1+64=65	64+343=407	512+1331=1843
1+125=126	64+512=576	512+1728=2240
1+216=217	64+729=793	512+2197=2709
1+343=344	64+1000=1064	512+2744=3256
1+512=513	64+1331=1395	512+3375=3887
1+729=730	64+1728=1792	512+4096=4608
1+1000=1001	64+2197=2261
1+1331=1332	64+2744=2808	729+1000=1729
1+1728=1729	64+3375=3439	729+1331=2060
1+2197=2198	64+4096=4160	729+1728=2457
1+2744=2755		729+2197=2926
1+3375=3376	125+216=341	729+2744=3473
1+4096=4097	125+343=468	729+3375=4104
	125+512=637	729+4096=4825
8+27=35	125+729=854
8+64=72	125+1000=1225	1000+1331=2331
8+125=133	125+1331=1456	1000+1728=2728
8+216=224	125+1728=1853	1000+2197=3197
8+343=351	125+2197=2322	1000+2744=3744
8+512=520	125+2744=2869	1000+3375=4375
8+729=737	125+3375=3500	1000+4096=5096
8+1000=1008	125+4096=4221
8+1331=1339		1331+1728=3059
8+1728=1736	216+343=559	1331+2197=3528
8+2197=2205	216+512=728	1331+2744=4075
8+2744=2752	216+729=945	1331+3375=4706
8+3375=3383	216+1000=1216	1331+4096=5427
8+4096=4104	216+1331=1547
	216+1728=1944	1728+2197=3925
27+64=91	216+2197=2413	1728+2744=4472
27+125=152	216+2744=2960	1728+3375=5103
27+216=243	216+3375=3591	1728+4096=5824
27+343=370	216+4096=4312
27+512=539		2197+2744=4941
27+729=756	343+512=855	2197+3375=5572
27+1000=1027	343+729=1072	2197+4096=6293
27+1331=1358	343+1000=1343
27+1728=1755	343+1331=1674	2744+3375=6119
27+2197=2224	343+1728=2071	2744+4096=6840
27+2744=2771	343+2197=2540
27+3375=3402	343+2744=3087	3375+4096=7471
27+4096=4123	343+3375=3718
	343+4096=4439

  

By seeing above table if we observe carefully sum of cubes 1729 appeared two time’s  which was shown in red colour.

1+1728=1729 and 1000+729=1729

Which can be re- written as
             1³+12³ =1729

And                    10³+9³=1729

And if we observe carefully another sum of cubes also repeated twice that is shown in blue colour.

8+4096=4104  and  729+3375=4106 and can be rewritten as

    2³ +16³=4104     and   

9³+15³=4104.