29 is the smallest positive whole number that cannot be made from the numbers, using each digit exactly once and using only addition, subtraction, multiplication, and division.[1] None of the first twenty-ninenatural numbers have more than two different prime factors (in other words, this is the longest such consecutive sequence; the firstsphenic number[2] or triprime,30 is the product of the first three primes2,3, and5). 29 is also,
It is also the largest prime factor of the smallest abundant number not divisible by the first even (of only one) and odd primes, 5391411025 = 52 × 7 × 11 × 13 × 17 × 19 × 23 × 29.[12] Both of these numbers are divisible by consecutive prime numbers ending in 29.
The15 and 290 theorems describes integer-quadratic matrices that describe allpositive integers, by the set of the first fifteen integers, or equivalently, the first two-hundred and ninety integers. Alternatively, a more precise version states that an integer quadratic matrix represents all positive integers when it contains the set oftwenty-nine integers between1 and290:[13][14]
The largest member 290 is the product between 29 and its index in thesequence of prime numbers,10.[15] The largest member in this sequence is also the twenty-fifth even,square-freesphenic number with three distinct prime numbers as factors,[16] and the fifteenth such that is prime (where in its case, 2 + 5 + 29 + 1 =37).[17][a]
The 29th dimension is the highest dimension forcompact hyperbolic Coxeter polytopes that are bounded by a fundamentalpolyhedron, and the highest dimension that holds arithmetic discrete groups of reflections withnoncompact unbounded fundamental polyhedra.[19]
^In this sequence,29 is the seventeenth indexed member, where the sum of the largest two members (203,290) is. Furthermore, 290 is the sum of the squares of divisors of17, or289 + 1.[18]
