Pierpont Prime

A Pierpont prime is a prime number of the form . The first few Pierpont primes are 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, ... (OEIS A005109).

A regular polygon of  sides can be constructed by ruler, compass and angle-trisector iff

where , , ...,  are distinct Pierpont primes and  (Gleason 1998).

The numbers of Pierpont primes less than , , ... are 4, 10, 18, 25, 32, 42, 50, 58, ... (OEIS A113420) and the number less than , , , , ... are 4, 10, 25, 58, 125, 250, 505, 1020, 2075, 4227, ... (OEIS A113412; Caldwell).

As of Apr. 2010, the largest known Pierpont prime is , which has  decimal digits (http://primes.utm.edu/primes/page.php?id=87449).

REFERENCES:

Caldwell, C. "Pierpont primes." primeform posting, Oct. 25, 2005. http://groups.yahoo.com/group/primeform/message/6588/.

Cox, D. A. and Shurman, J. "Geometry and Number Theory on Clovers." Amer. Math. Monthly 112, 682-704, 2005.

Gleason, A. M. "Angle Trisection, the Heptagon, and the Triskaidecagon." Amer. Math. Monthly 95, 185-194, 1988.

Guy, R. K. §A18 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.

Martin, G. E Geometric Constructions. New York: Springer, 1998.

Pierpont, J. "On an Undemonstrated Theorem of the Disquisitiones Arithmeticae." Bull. Amer. Math. Soc. 2, 77-83, 1895-1896.

Sloane, N. J. A. Sequences A005109/M0673, A113412, and A113420 in "The On-Line Encyclopedia of Integer Sequences."