Proth number

In number theory, a Proth number is a number of the form

: $P=k, 2^n+1$

where "k" is odd, "n" is a positive integer, and 2^"n" > "k". Proth numbers are named after the mathematician François Proth.

If a Proth number is prime, it is called Proth prime: Proth's theorem can be used to test the primality of a given Proth number.

Examples

The first Proth numbers are OEIS|id=A080075:

:"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

The first Proth primes are (OEIS2C|id=A080076)::3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857

ee also

*Sierpinski number

External links

*MathWorld|urlname=ProthNumber|title=Proth Number

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Proth number

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