的士數

第 n {\displaystyle n} 個的士數（Taxicab number），一般寫作 Ta ⁡ ( n ) {\displaystyle \operatorname {Ta} (n)} 或 Taxicab ⁡ ( n ) {\displaystyle \operatorname {Taxicab} (n)} ，定義為最小的數能以 n {\displaystyle n} 個不同的方法表示成兩個正立方數之和。1938年，G·H·哈代與愛德華·梅特蘭·賴特證明對於所有正整數 n {\displaystyle n} 這樣的數也存在。可是他們的證明對找尋的士數毫無幫助，截止現時，只找到6個的士數（ A011541）： Ta ⁡ ( 1 ) = 2 = 1 3 + 1 3 {\displaystyle {\begin{aligned}\operatorname {Ta} (1)=2&=1^{3}+1^{3}\end{aligned}}} Ta ⁡ ( 2 ) = 1729 = 1 3 + 12 3 = 9 3 + 10 3 {\displaystyle {\begin{aligned}\operatorname {Ta} (2)=1729&=1^{3}+12^{3}\\&=9^{3}+10^{3}\end{aligned}}} Ta ⁡ ( 3 ) = 87539319 = 167 3 + 436 3 = 228 3 + 423 3 = 255 3 + 414 3 {\displaystyle {\begin{aligned}\operatorname {Ta} (3)=87539319&=167^{3}+436^{3}\\&=228^{3}+423^{3}\\&=255^{3}+414^{3}\end{aligned}}} Ta ⁡ ( 4 ) = 6963472309248 = 2421 3 + 19083 3 = 5436 3 + 18948 3 = 10200 3 + 18072 3 = 13322 3 + 16630 3 {\displaystyle {\begin{aligned}\operatorname {Ta} (4)=6963472309248&=2421^{3}+19083^{3}\\&=5436^{3}+18948^{3}\\&=10200^{3}+18072^{3}\\&=13322^{3}+16630^{3}\end{aligned}}} Ta ⁡ ( 5 ) = 48988659276962496 = 38787 3 + 365757 3 = 107839 3 + 362753 3 = 205292 3 + 342952 3 = 221424 3 + 336588 3 = 231518 3 + 331954 3 {\displaystyle {\begin{aligned}\operatorname {Ta} (5)=48988659276962496&=38787^{3}+365757^{3}\\&=107839^{3}+362753^{3}\\&=205292^{3}+342952^{3}\\&=221424^{3}+336588^{3}\\&=231518^{3}+331954^{3}\end{aligned}}} Ta ⁡ ( 6 ) = 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 {\displaystyle {\begin{aligned}\operatorname {Ta} (6)=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}\end{aligned}}} Ta ⁡ ( 2 ) {\displaystyle \operatorname {Ta} (2)} 因為哈代和拉馬努金的故事而為人所知： 在 Ta ⁡ ( 2 ) {\displaystyle \operatorname {Ta} (2)} 之後，所有的的士數均用電腦來尋找。 David W. Wilson證明了 Ta ⁡ ( 6 ) ≤ 8230545258248091551205888 {\displaystyle \operatorname {Ta} (6)\leq 8230545258248091551205888} 。 1998年丹尼爾·朱利阿斯·伯恩斯坦（英语：Daniel J. Bernstein）證實 391909274215699968 ≥ Ta ⁡ ( 6 ) ≥ 10 18 {\displaystyle 391909274215699968\geq \operatorname {Ta} (6)\geq 10^{18}} 2002年Randall L. Rathbun證明 Ta ⁡ ( 6 ) ≤ 24153319581254312065344 {\displaystyle \operatorname {Ta} (6)\leq 24153319581254312065344} 2003年5月，Stuart Gascoigne確定 Ta ⁡ ( 6 ) > 6.8 × 10 19 {\displaystyle \operatorname {Ta} (6)>6.8\times 10^{19}} ，且Cristian S. Calude、Elena Calude及Michael J. Dinneen顯示 Ta ⁡ ( 6 ) = 24153319581254312065344 {\displaystyle \operatorname {Ta} (6)=24153319581254312065344} 的機會大於99%。 G. H. Hardy和E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412. J. Leech, Some Solutions of Diophantine Equations, Proc. Cambridge Phil. Soc. 53, 778-780, 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equation s = x3 + y3 = z3 + w3 = u3 + v3 = m3 + n3, Bull. Inst. Math. Appl., 27(1991) 155-157; MR 92i:11134, online （页面存档备份，存于互联网档案馆） David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online （页面存档备份，存于互联网档案馆） D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d), Mathematics of Computation 70, 233 (2000), 389—394. C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), p. 1196-1203 一般化的士數：多個多次冪之和 士的數：兩個不論正負的立方數之和 三立方数和 A 2002 post to the Number Theory mailing list by Randall L. Rathbun （页面存档备份，存于互联网档案馆） Grime, James; Bowley, Roger. Haran, Brady , 编. 1729: Taxi Cab Number or Hardy-Ramanujan Number. Numberphile. [2020-10-25]. （原始内容存档于2017-03-06）.  Taxicab and other maths at Euler （页面存档备份，存于互联网档案馆） Singh, Simon. Haran, Brady , 编. Taxicab Numbers in Futurama. Numberphile. [2020-10-25]. （原始内容存档于2015-12-03）.