Theorem of Proth

In Mathematical, the theorem of Proth in Théorie of the numbers is a Test of primality for the Nombres of Proth.

This theorem states that for a number of Proth p , therefore form K 2^N + 1 with K a naturalness and K < 2^N, if there exists an entirety has such as:

$a^ {(p-1) /2} \ equiv -1 \ pmod {p} \, \!$

then p is first.

This test is practical because if p is first, a has chosen has approximately 50% of chances to prove the primality of p . Moreover it is remarkably useful to show the conjecture of Sierpinski.

Numerical examples

The first seven numbers of Proth correspond to:

P ₀ = 2¹ + 1 = 3

P ₁ = 2² + 1 = 5

P ₂ = 2³ + 1 = 9

P ₃ = 3 × 2² + 1 = 13

P ₄ = 2⁴ + 1 = 17

P ₅ = 3 × 2³ + 1 = 25

P ₆ = 2⁵ + 1 = 33

Examples of the theorem:

For p = 3,2¹ + 1 = 3 what is divisible by 3; thus 3 is first.
For p = 5,3² + 1 = 10 what is divisible by 5; thus 5 is first.
For p = 13,5⁶ + 1 = 15626 what is divisible by 13; thus 13 are first.
For p = 9, which is not first, there does not exist has such as has ⁴ + 1 is divisible by 9.

History

the mathematician François Proth (1852 - 1879) discovered the theorem in 1878.

See too

External bonds

Mathworld

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