Sunday, February 18, 2018

[oebfsidv] Taxicab numbers

There are many ways to generalize the Hardy-Ramanujan number 1729:

The {a}th number which can be written in at least {b} different ways as the sum of {c} numbers all of the form {d}^{e}.

The {d}s could be restricted to positive and/or relatively prime.  The form {d}^{e} could be relaxed to any form, e.g., triangular numbers (Gauss's Eureka theorem).

The most famous generalization is the sequence that is a function of b, restricting a=1 c=2 e=3 and d=non-negative.

I like the sequence that is a function of a, b=2 c=2 e=3 and d=unrestricted.  This is A051347.  Allowing negative numbers seems appropriate for the cubing operation as the range of cubing extends to negative numbers (unlike for squaring).  If cubing can do it, let it.

Waring's problem is about b=1, and its most narrow form restricts to nonnegative d.   But how many signed cubes (what value of c) does it take to express any sufficiently large number?  Probably 4 (same as with positive-only, though the "sufficiently large" threshold might be different).  Coincidentally, this is the same number of squares needed, by Lagrange's Four-Square Theorem.

Incidentally, it is not too difficult to notice that 1729 can be expressed as the sum of positive cubes in 2 different ways.  If one happens to have cubes memorized up to 12^3=1728, then one has probably also noticed that 9^3=729 differs from 12^3 by almost exactly 1000, and 1000 is also a cube.  From this, it is easy to realize 12^3+1^3 = 9^3+10^3.

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