29 is the tenth
prime number.
Integer properties 29 is the fifth
primorial prime, like its
twin prime 31. 29 is the smallest positive whole number that cannot be made from the numbers \{1, 2, 3, 4\}, using each digit exactly once and using only addition, subtraction, multiplication, and division. None of the first twenty-nine
natural numbers have more than two different prime factors (in other words, this is the longest such consecutive sequence; the first
sphenic number or triprime,
30 is the product of the first three primes
2,
3, and
5). 29 is also, • the sum of three consecutive
squares,
22 +
32 +
42. • the sixth
Sophie Germain prime. • a
Lucas prime, a
Pell prime, and a
tetranacci number. • an
Eisenstein prime with no imaginary part and real part of the form 3n − 1. • a
Markov number, appearing in the solutions to
x +
y +
z = 3
xyz: {2, 5, 29}, {2, 29, 169}, {5, 29, 433}, {29, 169, 14701}, etc. • a
Perrin number, preceded in the sequence by 12, 17, 22. • the number of
pentacubes if reflections are considered distinct. • the tenth
supersingular prime. On the other hand, 29 represents the sum of the first cluster of consecutive
semiprimes with distinct
prime factors (
14,
15). These two numbers are the only numbers whose
arithmetic mean of divisors is the first
perfect number and
unitary perfect number,
6 (that is also the smallest semiprime with distinct factors). The pair (14, 15) is also the first
floor and ceiling values of
imaginary parts of non-trivial zeroes in the
Riemann zeta function, \zeta. 29 is the largest
prime factor of the smallest number with an
abundancy index of 3, :1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29 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. Both of these numbers are divisible by consecutive prime numbers ending in 29.
15 and 290 theorems The
15 and 290 theorems describes integer-quadratic matrices that describe all
positive 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 of
twenty-nine integers between
1 and
290: :\{1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290\} The largest member 290 is the product between 29 and its index in the
sequence of prime numbers,
10. The largest member in this sequence is also the twenty-fifth even,
square-free
sphenic number with three distinct prime numbers p \times q \times r as factors, and the fifteenth such that p + q + r + 1 is prime (where in its case, 2 + 5 + 29 + 1 =
37).
Dimensional spaces The 29th dimension is the highest dimension for
compact hyperbolic Coxeter polytopes that are bounded by a fundamental
polyhedron, and the highest dimension that holds arithmetic discrete groups of reflections with
noncompact unbounded fundamental polyhedra. == Notes ==