Pierpont prime
A Pierpont prime is a prime number of the form formula_1 for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections, they are also regular polygons that can be constructed using ruler, compass, and angle trisector, they are also regular polygons that can be constructed using paper folding. A Pierpont prime with is of the form formula_2, and is therefore a Fermat prime (unless ). If is positive  then must also be positive (because formula_3 would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Piermont primes all have the form , when is a positive integer (except for 2, when ). The first few Pierpont primes are: , the largest known Pierpont prime is 3×2 + 1, which has 4,939,547 decimal digits. Distribution. Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 10, 65 less than 10, 157 less than 10, and 795 less than 10. There are few restrictions from  algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among -digit numbers of the correct form formula_4, the fraction of these that are prime should be proportional to , a similar proportion as the proportion of prime numbers among all -digit numbers. As there are formula_5 numbers of the correct form in this range, there should be formula_6 Pierpont primes. Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately Pierpont  primes up to . According to Gleason's conjecture there are formula_7 Pierpont primes smaller than "N", as opposed to the smaller conjectural number formula_8 of Mersenne primes in that range. Primality testing. When formula_9, the primality of formula_10 can be tested by Proth's theorem. On the other hand, when formula_11 alternative primality tests for formula_12 are possible based on the factorization of formula_13 as a small even number multiplied by a large power of 3. Polygon construction. In the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown  that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of sides to be formed, as long as and is of the form , where is a product of distinct Pierpont primes. This is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector. Regular polygons which can be constructed with only compass and straightedge (constructible polygons) are the special case where and is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In  1895, James Pierpont studied the same class of regular polygons; his work is what gives the name to the Pierpont primes. Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular -gons that can be constructed with these operations are the ones such that the totient of is 3-smooth. Since the totient of a prime is formed by subtracting one from it, the primes for which Pierpont's construction works are exactly the Pierpont primes. However, Pierpont did not  describe the form of the composite numbers with 3-smooth totients. As Gleason later showed, these numbers are exactly the ones of the form given above. The smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the hendecagon is the first regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections). All other regular with can be constructed with compass, straightedge and trisector. Generalization. A Pierpont prime of the second kind is a prime number of the form 23 − 1. These numbers are The largest known primes  of this type are Mersenne primes; currently the largest known is formula_14 (24,862,048 decimal digits). The largest known Pierpont prime of the second kind that is not a Mersenne prime is formula_15. A generalized Pierpont prime is a prime of the form formula_16 with "k" fixed primes "p" < "p" < "p" < ... < "p". A generalized Pierpont prime of the second kind is a prime of the form formula_17 with "k" fixed primes "p" < "p" < "p" < ... < "p". Since all primes greater than 2 are odd, in both kinds "p" must be 2. The sequences of  such primes in the OEIS are: