Taxicab number

In mathematics, a taxicab number is a special kind of number. It is the smallest number that can be written as the sum of two positive integer cubes in different ways. The most famous example is 1729.

This number gets its name from a story about two mathematicians, G. H. Hardy and Srinivasa Ramanujan. Hardy visited Ramanujan and mentioned the taxi number 1729. Ramanujan said it was interesting because it can be written as the sum of two cubes in two different ways.

Taxicab numbers help mathematicians explore fun puzzles about numbers and their properties.

History and definition

The number 1729 is special because it can be written as the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³. This number was first noticed in 1657 and later became famous because of a story about the mathematicians Srinivasa Ramanujan.

Mathematicians have used computers to find more numbers like 1729. These numbers are called taxicab numbers, and each one can be written as the sum of two cubes in a certain number of different ways. For example, Ta(3) was found in 1957, and Ta(4) was found in 1989.

Known taxicab numbers

Mathematicians have found six special numbers called taxicab numbers. These numbers can be written as the sum of two cubes in different ways. The smallest taxicab number is 2, which can be written as 1³ + 1³. The next one is 1729, also known as the Hardy-Ramanujan number, and it can be written as 1³ + 12³ or 9³ + 10³. There are also larger taxicab numbers, like 87,539,319 and even bigger ones.

Upper bounds for taxicab numbers

For the following taxicab numbers, we know the highest possible values they could have:

Ta(7) ≤ 24885189317885898975235988544 = 58798362³ + 2919526806³ = 309481473³ + 2918375103³ = 459531128³ + 2915734948³ = 860447381³ + 2894406187³ = 1638024868³ + 2736414008³ = 1766742096³ + 2685635652³ = 1847282122³ + 2648660966³

Ta(8) ≤ 50974398750539071400590819921724352 = 7467391974³ + 370779904362³ = 39304147071³ + 370633638081³ = 58360453256³ + 370298338396³ = 109276817387³ + 367589585749³ = 208029158236³ + 347524579016³ = 224376246192³ + 341075727804³ = 234604829494³ + 336379942682³ = 288873662876³ + 299512063576³

Ta(9) ≤ 136897813798023990395783317207361432493888 = 1037967484386³ + 51538406706318³ = 4076877805588³ + 51530042142656³ = 5463276442869³ + 51518075693259³ = 8112103002584³ + 51471469037044³ = 15189477616793³ + 51094952419111³ = 28916052994804³ + 48305916483224³ = 31188298220688³ + 47409526164756³ = 32610071299666³ + 46756812032798³ = 40153439139764³ + 41632176837064³

Ta(10) ≤ 7335345315241855602572782233444632535674275447104 = 391313741613522³ + 19429979328281886³ = 904069333568884³ + 19429379778270560³ = 1536982932706676³ + 19426825887781312³ = 2059655218961613³ + 19422314536358643³ = 3058262831974168³ + 19404743826965588³ = 5726433061530961³ + 19262797062004847³ = 10901351979041108³ + 18211330514175448³ = 11757988429199376³ + 17873391364113012³ = 12293996879974082³ + 17627318136364846³ = 15137846555691028³ + 15695330667573128³

Ta(11) ≤ 87039729655193781808322993393446581825405320183232000 = 5134510178400057³ + 443171971973855943³ = 27089483598685872³ + 443138459854855128³ = 127174000598779680³ + 439653507772479000³ = 138573856797762960³ + 438609133406051160³ = 204623083640747772³ + 428126038425768228³ = 209891877907138700³ + 426887616463852180³ = 212424209933109720³ + 426267111265435440³ = 299032406381730840³ + 392138457234189120³ = 301539992238035460³ + 390662458762053660³ = 309479752750029680³ + 385744811881975000³ = 316469686016945240³ + 381087194739069520³

Ta(12) ≤ 16119148654034302034428760115512552827992287460693283776000 = 292667080168803249³ + 25260802402509788751³ = 771180546485662040³ + 25260575914339118080³ = 1544100565125094704³ + 25258892211726742296³ = 7248918034130441760³ + 25060249943031303000³ = 7898709837472488720³ + 25000720604144916120³ = 11663515767522623004³ + 24403184190268788996³ = 11963837040706905900³ + 24332594138439574260³ = 12108179966187254040³ + 24297225342129820080³ = 17044847163758657880³ + 22351892062348779840³ = 17187779557568021220³ + 22267760149437058620³ = 17640345906751691760³ + 21987454277272575000³ = 18038772102965878680³ + 21721970100126962640³

Cubefree taxicab numbers

A special kind of taxicab number must be cubefree, meaning it cannot be divided by any cube larger than 1³. For these numbers, the two cubes that add up to the number must also share no common factors.

Only the first two taxicab numbers are cubefree. In 1981, a mathematician named Paul Vojta found the smallest cubefree taxicab number that can be written as the sum of two cubes in three different ways. Even bigger numbers with more ways to be written as the sum of two cubes have been found since then.