4,104 shares the exact same distinctive property that made 1,729 famous — expressibility as the sum of two positive cubes in two genuinely different ways — and it holds the honor of being the smallest number after 1,729 with this property: 4,104 = 2³ + 16³ (8 + 4,096) and also 9³ + 15³ (729 + 3,375), both landing on exactly 4,104, with no smaller number between 1,729 and 4,104 sharing this same dual-representation property.
Numbers with this property are called taxicab numbers, named after the anecdote involving 1,729 and the mathematicians G. H. Hardy and Srinivasa Ramanujan (covered in full on this site's own page for 1,729). The formal notation labels them by how many distinct ways they can be written as a sum of two positive cubes: Ta(1) = 2 (the trivial case, 1³+1³, the smallest sum of two positive cubes at all, though only expressible one way), Ta(2) = 1,729 (the smallest number with *two* distinct such representations), and — while 4,104 shares Ramanujan's exact property — the formal Ta(2) designation belongs specifically to 1,729 as the smallest example; 4,104 is simply the next number in ascending order sharing that same "two distinct ways" property, making it a natural, closely related companion fact rather than a separately numbered taxicab record itself.
Finding numbers like 4,104 by hand is a genuinely more demanding search than checking a single candidate for primality or perfection, precisely because it requires searching across pairs of cube combinations rather than checking a single number's own properties in isolation — for any candidate total, you'd need to check every pair of cubes below it to see whether two genuinely different pairs land on the same sum, which is part of why these numbers, while conceptually simple to state, are much more naturally suited to computer search than to hand calculation once you move beyond the first couple of known examples.
4,104 factors, as a plain integer, into 2³ × 3³ × 19 — a composite number with sixteen divisors total, following from the standard divisor-counting formula applied to its prime factorization ((3+1)×(3+1)×(1+1)=16). It's worth noting this ordinary factorization has no direct, visible connection to its sum-of-two-cubes property — unlike, say, a perfect number's factorization, which directly encodes its Mersenne-prime structure, a taxicab number's defining property is a fact about which specific cube pairs happen to sum to it, not something legible from its prime factorization at a glance.
The taxicab number sequence continues well beyond 4,104 — the third, Ta(3) = 87,539,319, is expressible as a sum of two cubes in three distinct ways, and each subsequent term in the sequence grows dramatically larger and harder to locate, a pattern of rapidly widening gaps that echoes the similarly fast-growing spacing seen among perfect numbers and other rare number-theoretic categories covered elsewhere on this site.
It's worth being precise about a small but genuinely important distinction some casual references blur: not every number expressible as a sum of two cubes in two ways is itself called a "taxicab number" in the strictest formal sense — the term, and its Ta(n) numbering, specifically identifies the *smallest* number with exactly n such representations. Larger numbers sharing 4,104's exact "two distinct cube-sum pairs" property exist as well, but only 1,729 earns the formal Ta(2) designation, since it's the smallest; 4,104 is more precisely described as the next number after 1,729 sharing that same underlying property, a genuine and useful distinction for anyone reading further into taxicab-number literature.
Both of 4,104's cube-sum pairs are worth double-checking directly, since they're small enough to verify without a calculator: 2³ = 8, 16³ = 4,096, and 8 + 4,096 = 4,104 exactly; separately, 9³ = 729, 15³ = 3,375, and 729 + 3,375 = 4,104 as well — two genuinely different pairs of cubes, using entirely different base numbers, landing on the identical total, which is the whole striking point of the taxicab-number property in the first place — a fact easy to state but genuinely satisfying to verify personally rather than simply take on faith, and a small, self-contained example of exactly the kind of hands-on checking this site's approach to number facts consistently favors — arithmetic worth trusting because it's been shown, not merely asserted.