29 (number)

Mathematics

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 = 3xyz: {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

References

References

1. "Sloane's A060315". OEIS Foundation.
2. {{Cite OEIS. A007304. sphenic numbers
3. "Sloane's A005384 : Sophie Germain primes". OEIS Foundation.
4. "Sloane's A005479 : Prime Lucas numbers". OEIS Foundation.
5. "Sloane's A086383 : Primes found among the denominators of the continued fraction rational approximations to sqrt(2)". OEIS Foundation.
6. "Sloane's A000078 : Tetranacci numbers". OEIS Foundation.
7. "Sloane's A001608 : Perrin sequence". OEIS Foundation.
8. "Sloane's A002267 : The 15 supersingular primes". OEIS Foundation.
9. {{Cite OEIS. A001358. Semiprimes (or biprimes): products of two primes.
10. {{Cite OEIS. A003601. Numbers j such that the average of the divisors of j is an integer.
11. {{Cite OEIS. A102187. Arithmetic means of divisors of arithmetic numbers
12. {{Cite OEIS. A047802. Least odd number k such that sigma(k)/k is greater than or equal to n.
13. (2007). "Number Theory Volume I: Tools and Diophantine Equations". [[Springer Science+Business Media.
14. {{Cite OEIS. A030051. Numbers from the 290-theorem.
15. {{Cite OEIS. A033286. a(n) as n * prime(n).
16. {{Cite OEIS. A075819. Even squarefree numbers with exactly 3 prime factors.
17. {{Cite OEIS. A291446. Squarefree triprimes of the form pqr such that p + q + r + 1 is prime.
18. {{Cite OEIS. A001157. a(n) as sigma_2(n): sum of squares of divisors of n.
19. Vinberg, E.B.. (1981). "Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension". [[Springer Science+Business Media.