3,435 has a genuinely strange self-referential property: raise each of its own digits to a power equal to itself, and add the results together, and you get 3,435 right back. The digits are 3, 4, 3, 5 — so the calculation is 3³ + 4⁴ + 3³ + 5⁵ = 27 + 256 + 27 + 3,125 = 3,435. Each digit is raised to the power of its own value (the first 3 to the 3rd power, the 4 to the 4th power, the second 3 to the 3rd power again, and the 5 to the 5th power), and the resulting sum reproduces the original number exactly — a number that, in a very literal sense, is built entirely and exactly out of itself.
Numbers with this property are called Münchhausen numbers, named after Baron Münchhausen, the fictional tall-tale character (based loosely on a real 18th-century German nobleman, later wildly embellished in popular literature) famous for claiming, among other outlandish feats, to have pulled himself and his horse out of a swamp by his own hair — a fitting namesake for a number that, mathematically, pulls itself into existence from its own digits. The term itself is comparatively recent, coined by the Dutch mathematician and software developer Daan van Berkel in a 2009 paper specifically studying this digit property.
3,435 is genuinely rare — in base 10, using the standard convention that 0⁰=1 (a convention mathematicians adopt specifically for this kind of calculation, since without it the property becomes undefined for any number containing a 0 digit), there are only two nontrivial Münchhausen numbers: 1 (trivially, since 1¹=1) and 3,435 itself. No other base-10 number, however large, has ever been found or proven to share this exact property, and it's provable (through a fairly short bounding argument on how large the digit-power sum can possibly grow relative to the number of digits) that no larger one exists in base 10 — meaning 3,435 isn't just the largest currently known example, it's mathematically the largest one that can ever exist under this exact definition.
It's worth being precise about the "nontrivial" qualifier: some sources also count 0 itself, under the convention 0⁰=1 applied to a single-digit number, though this is a genuinely contested edge case among people who study the property, since it depends entirely on which convention for 0⁰ (a notoriously ambiguous expression in mathematics generally) a given source adopts.
As a plain integer, 3,435 factors as 3 × 5 × 229 — a straightforward composite number with eight divisors (1, 3, 5, 15, 229, 687, 1145, 3435), following directly from its three distinct prime factors. Its entire mathematical fame rests on the Münchhausen property described above rather than on this comparatively ordinary factorization — a reminder that a number's most interesting feature often lies in a property specific to how its digits relate to the number as a whole, quite separate from its divisor structure.
Münchhausen numbers exist in other number bases too, and their specific values differ meaningfully from base to base because the whole property is defined in terms of digit values, which are themselves base-dependent — a number's digits in base 10 bear no fixed relationship to its digits in, say, base 7. In base 4, for instance, the equivalent search (using the convention 0⁰=1) turns up 3,110,220 (in base 4 notation) as a genuine base-4 Münchhausen number, a completely different-looking value from 3,435 despite representing the identical underlying mathematical concept applied to a different counting system — a reminder that "how a digit relates to the whole number" is fundamentally a base-specific question, not a universal property of the number's abstract magnitude.
Daan van Berkel's original 2009 paper coining the term "Münchhausen number" also formally established the proof that 3,435 is the largest possible nontrivial example in base 10 — not merely the largest one anyone has happened to find through exhaustive search, but a value mathematically guaranteed to be the ceiling, since the paper's bounding argument (comparing the maximum possible digit-power sum against the minimum value a number with that many digits must have) rules out any larger example ever existing, however far a computer search might eventually extend — a rare instance of a recreational-mathematics curiosity being fully, definitively closed rather than left as an ongoing open search.