MATHS

Numbers which represent the sum of two cubes in two different ways are call taxicab numbers, from a historical conversation between the mathematicians G.H. Hardy and Srinivasa Ramanujan. (see, for example: http://pi.lacim.uqam.ca/eng/problem_en.html)

A taxicab function of order n represents the smallest number which can be can be represented by the sum of two cubes in n different ways.
For example, Taxicab(2) = 1729.

What your question asks is the next (higher) number that has the same properties as Taxicab(2).

Here are the first 30 numbers that have the properties of Taxicab(2), i.e. each number can be represented by the sum of two cubes in two different ways.
The numbers tobe cubed are shown in parentheses.

Taxicab(2)[1]=1729=(12,1),(10,9)
Taxicab(2)[2]=4104=(16,2),(15,9)
Taxicab(2)[3]=13832=(24,2),(20,18)
Taxicab(2)[4]=20683=(27,10),(24,19)
Taxicab(2)[5]=32832=(32,4),(30,18)
Taxicab(2)[6]=39312=(34,2),(33,15)
Taxicab(2)[7]=40033=(34,9),(33,16)
Taxicab(2)[8]=46683=(36,3),(30,27)
Taxicab(2)[9]=64232=(39,17),(36,26)
Taxicab(2)[10]=65728=(40,12),(33,31)
Taxicab(2)[11]=110656=(48,4),(40,36)
Taxicab(2)[12]=110808=(48,6),(45,27)
Taxicab(2)[13]=134379=(51,12),(43,38)
Taxicab(2)[14]=149389=(53,8),(50,29)
Taxicab(2)[15]=165464=(54,20),(48,38)
Taxicab(2)[16]=171288=(55,17),(54,24)
Taxicab(2)[17]=195841=(58,9),(57,22)
Taxicab(2)[18]=216027=(60,3),(59,22)
Taxicab(2)[19]=216125=(60,5),(50,45)
Taxicab(2)[20]=262656=(64,8),(60,36)
Taxicab(2)[21]=327763=(67,30),(58,51)
Taxicab(2)[22]=314496=(68,4),(66,30)
Taxicab(2)[23]=320264=(68,18),(66,32)
Taxicab(2)[24]=402597=(69,42),(61,56)
Taxicab(2)[25]=373464=(72,6),(60,54)
Taxicab(2)[26]=439101=(76,5),(69,48)
Taxicab(2)[27]=443889=(76,17),(73,38)
Taxicab(2)[28]=513856=(78,34),(72,52)
Taxicab(2)[29]=513000=(80,10),(75,45)
Taxicab(2)[30]=515375=(80,15),(71,54)