Prime

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

prime(n) is a nonanalytical function which is defined only for positive integers (SuperPlus[ℕ]).

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Other series representations

Identities

Functional identities

Products

Finite products

Infinite products

Regularized products

Operations

Limit operation

Representations through equivalent functions

With related functions

Inequalities

Other identities

Congruence properties

Various Conjectures

Theorems

Fermats' little theorem

(a^(p-1))≡1(mod p)/; p∈ℙ∧(a/p)∉ℕ.

Representation of positive integers through primes (fundamental theorem of arithmetic)

Every positive integer n can be uniquely represented in the form  n==(∏_(k=1)^r)((Subscript[p, k])^(Subscript[n, k])),(Subscript[p, k])∈ℙ∧(Subscript[n, k])∈(SuperPlus[ℕ]),(Subscript[p, k])<(Subscript[p, k+1]).

Bertrand's hypothesis

For all n∈ℕ,n>1 there is at least one prime between n and 2n.

One linearly independent set

The set (((Subscript[p, 1]^(Subscript[α, 1]))(Subscript[p, 2]^(Subscript[α, 2]))… (Subscript[p, n]^(Subscript[α, n])))^(1/k)) , (Subscript[p, i])∈ℙ, k∈ℕ, 0≤(Subscript[α, i])≤k is linearly independent over  ℤ.

Giuga's conjecture

If an integer n satisfies (∑_(k=1)^(n-1))(k^(n-1))==-1 mod n, then n is prime.

One linearly independent set

The set of primes  ℙ coincides with the set of positive values of the following polynomial:

z(k+2) (-(((a i-i+k-l+1))^2)-(((((a^2)-1) (l^2)-(m^2)+1))^2)(((16 (k+2) (((n+1))^2) (((k+1))^3)-(f^2)+1))^2)-
((((((a+1))^2) (e+2) (e^3)-(o^2)+1))^2)-(((-m+l (a-n-1)+b (-(n^2)+2 a n-2 n+2 a-2)+p))^2)-
(((l+n+v-y))^2)-(((q+(-(p^2)-2 p+2 a+2 a p-2) s-x+(a-p-1) y))^2)-(((-(x^2)+((a^2)-1) (y^2)+1))^2)-
(((16 ((a^2)-1) (r^2) (y^4)-(u^2)+1))^2)-(((-(((c u+x))^2)+((((((u^2)-a) (u^2)+a))^2)-1) (((n+4 d y))^2)+1))^2)-
 (((h+(h+j) (k g+2 g+k+1)-z))^2)-
(((-e+2 n+p+q+z))^2)(((-m p+l (a-p) p+(-(p^2)+2 a p-1) t+z))^2)-(((h+j-q+w z))^2)+1),

when a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,x,y,w, z∈ℕ .

History

– Euclid
– P. J. Mollerup (1922)

The prime numbers are encountered often in mathematics and the natural sciences.

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