Número taxicab

Fotografia de S. A. Ramanujan.

Es diu que un número és l'enèsim número taxicab si és el menor número que es pot descompondre com a n sumes diferents de dos cubs positius. El nom d'aquests nombres prové d'una anècdota entre els matemàtics G. H. Hardy i S. A. Ramanujan (vegeu número de Hardy-Ramanujan).

Els nombres taxicab coneguts són aquests:

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}}}

Referències

  • G. H. Hardy and 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. Camb. 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 equations = x3 + y3 = z3 + w3 = u3 + v3 = m³ + n3, Bull. Inst. Math. Appl., 27(1991) 155–157; MR1125858, online.
  • David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online. (Wilson was unaware of J. A. Dardis' prior discovery of Ta(5) in 1994 when he wrote this.)
  • 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

Vegeu també