Prime Degree

In mathematics, a prime number is a prime raised to a positive integer .

Content

Examples

The numbers 5 = 5 ¹ , 9 = 3 ² and 16 = 2 ⁴ are powers of primes, while 6 = 2 × 3, 15 = 3 × 5 and 36 = 6 ² = 2 ² × 3 ² are not.

Twenty smallest degrees of prime numbers ^[1] :

2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , 17 , 19 , 23 , 25 , 27 , 29 , 31 , 32 , 37 , 41 , ...

Properties

Algebraic properties

Each power of a prime number is divided by only one prime number.
The density distribution of powers of primes is asymptotically equivalent $\pi (x)$ ${\ displaystyle \ pi (x)}$ - densities of primes up to $O({\sqrt {x}})$ ${\ displaystyle O ({\ sqrt {x}})}$ .
Any power of a prime (except power of 2) has a primitive root . So, the multiplicative group of integers modulo p ⁿ (or, equivalently, the group of units of the ring Z / p ⁿ Z ) is cyclic .
The number of elements of a finite field is always the degree of a prime number and vice versa; any degree of a prime is the number of elements of a certain finite field (unique up to isomorphism ).

Combinatorial Properties

The property of powers of a prime number, often used in analytic number theory , is that the set of powers of primes that are not prime is in the sense that an infinite sum of inverse quantities converges , although the set of primes is a large set.

Divisibility Properties

The Euler function ( φ ) and the sigma functions ( σ ₀ ) and ( σ ₁ ) of the degree of a prime can be calculated by the formulas:

\phi (p^{n})=p^{n-1}\phi (p)=p^{n-1}(p-1)=p^{n}-p^{n-1}=p^{n}\left(1-{\frac {1}{p}}\right),

{\ displaystyle \ phi (p ^ {n}) = p ^ {n-1} \ phi (p) = p ^ {n-1} (p-1) = p ^ {n} -p ^ {n- 1} = p ^ {n} \ left (1 - {\ frac {1} {p}} \ right),}

\sigma _{0}(p^{n})=\sum _{j=0}^{n}p^{0*j}=\sum _{j=0}^{n}1=n+1,

{\ displaystyle \ sigma _ {0} (p ^ {n}) = \ sum _ {j = 0} ^ {n} p ^ {0 * j} = \ sum _ {j = 0} ^ {n} 1 = n + 1,}

\sigma _{1}(p^{n})=\sum _{j=0}^{n}p^{1*j}=\sum _{j=0}^{n}p^{j}={\frac {p^{n+1}-1}{p-1}}.

{\ displaystyle \ sigma _ {1} (p ^ {n}) = \ sum _ {j = 0} ^ {n} p ^ {1 * j} = \ sum _ {j = 0} ^ {n} p ^ {j} = {\ frac {p ^ {n + 1} -1} {p-1}}.}

All degrees of primes are insufficient numbers . The degree of prime p ⁿ is n - . It is not known whether the powers of primes p ^{n can} be friendly numbers . If such numbers exist, then p ⁿ must be greater than 10 ¹⁵⁰⁰ and n must be greater than 1400.

Notes

↑ Sequence A000961 in OEIS : Degrees of Prime Numbers = Powers of primes

Literature

Jones, Gareth A. and Jones, J. Mary. Springer-Verlag. Elementary Number Theory. - London: Limited, 1998.

[1] Sequence A000961 in OEIS : Degrees of Prime Numbers = Powers of primes