Proth number

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Proth number

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After French mathematician François Proth (1852-1879).

Proth number (plural Proth numbers)

1. (number theory) Any number of the form k·2ⁿ + 1, where k is odd, n is a positive integer, and 2ⁿ > k.
   • 2006, B. Grégoire, L. Théry, B. Werner, A Computational Approach to Pocklington Certificates, Masami Hagiya, Philip Wadler (editors), Functional and Logic Programming: 8th International Symposium, Proceedings, Springer, LNCS 3945, page 109,

     To generate Pocklington certificates for Proth number we add a new entry to the oracle: pocklington -proth k p.

   • 2016, Abhijit Das, Computational Number Theory‎, Taylor & Francis (CRC Press / Chapman & Hall), page 295:

     Suppose that a Proth number ${\displaystyle n=k2^{r}+1}$ satisfies the condition that ${\displaystyle a^{(n-1)/2}\equiv -1{\pmod {n}}}$ for some integer ${\displaystyle a}$. Prove that ${\displaystyle n}$ is prime.

   • 2014, Adam Spencer, Adam Spencer's Big Book of Numbers, Brio Books, page 388:

     If a Proth number is prime, we call it a Proth prime.

• Cullen number
• Proth prime

• Sierpinski number

• Sequence A080075 on the On-Line Encyclopedia of Integer Sequences