{"history":[],"article":{"id":"aa458e32-bf1b-4044-a943-bd82cd70b7ab","title":"Pierpont prime","slug":"pierpont-prime","content":":::summary\n**Named After:** [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician) \n**Terms Number:** Thousands \n**Con Number:** Infinite \n**Parentsequence:** Pierpont number \n**First Terms:** 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889 \n**Largest Known Term:** 81 × 220,498,148 + 1 \n**Oeis:** A005109 \n:::\n\n# \nIn [number theory](https://bliptext.com/articles/number-theory), a **Pierpont prime** is a [prime number](https://bliptext.com/articles/prime-number) of the form\nfor some nonnegative [integers](https://bliptext.com/articles/integer) u and v. That is, they are the prime numbers p for which\np − 1\nis [3-smooth](https://bliptext.com/articles/smooth-number).\nThey are named after the mathematician [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician), who used them to characterize the [regular polygons](https://bliptext.com/articles/regular-polygon) that can be constructed using [conic sections](https://bliptext.com/articles/conic-section).\nThe same characterization applies to polygons that can be constructed using ruler, compass, and [angle trisector](https://bliptext.com/articles/angle-trisection), or using [paper folding](https://bliptext.com/articles/mathematics-of-paper-folding).\nExcept for 2 and the [Fermat primes](https://bliptext.com/articles/fermat-prime), every Pierpont prime must be 1 modulo 6.\nThe first few Pierpont primes are:\nIt has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.\n\n# Distribution\nA Pierpont prime with\nv = 0\nis of the form 2^u+1, and is therefore a [Fermat prime](https://bliptext.com/articles/fermat-prime) (unless\nu = 0\n). If v is [positive](https://bliptext.com/articles/positive-number) then u must also be [positive](https://bliptext.com/articles/positive-number) (because 3^v+1 would be an [even number](https://bliptext.com/articles/even-number) greater than 2 and therefore not prime), and therefore the non-Fermat Pierpont primes all have the form\n6k + 1\n, when k is a positive integer (except for 2, when\nu = v = 0\n).\nEmpirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 106, 65 less than 109, 157 less than 1020, and 795 less than 10100.\nThere are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the [Mersenne prime](https://bliptext.com/articles/mersenne-prime) condition that the exponent must be prime.\nThus, it is expected that among n-digit numbers of the correct form, the fraction of these that are prime should be proportional to\n1/n\n, a similar proportion as the proportion of prime numbers among all n-digit numbers.\nAs there are numbers of the correct form in this range, there should be \\Theta(n) Pierpont primes.\n[Andrew M. Gleason](https://bliptext.com/articles/andrew-m-gleason) made this reasoning explicit, [conjecturing](https://bliptext.com/articles/conjecture) there are infinitely many Pierpont primes, and more specifically that there should be approximately\n9n\nPierpont primes up to\n10n\n. According to Gleason's conjecture there are Pierpont primes smaller than N, as opposed to the smaller conjectural number of Mersenne primes in that range.\n# [Distribution of the exponents for the smaller Pierpont primes | upload.wikimedia.org/wikipedia/commons/a/a0/Pierpont///exponent///distribution.png]\n\n# Primality testing\nWhen 2^u > 3^v, is a [Proth number](https://bliptext.com/articles/proth-number) and thus its primality can be tested by [Proth's theorem](https://bliptext.com/articles/proth-s-theorem).\nOn the other hand, when 2^u < 3^v alternative primality tests for are possible based on the factorization of M-1 as a small even number multiplied by a large [power of 3](https://bliptext.com/articles/power-of-3).\n\n# Pierpont primes found as factors of Fermat numbers\nAs part of the ongoing worldwide search for factors of [Fermat numbers](https://bliptext.com/articles/fermat-number), some Pierpont primes have been announced as factors.\nThe following table gives values of m, k, and n such that\nThe left-hand side is a Fermat number; the right-hand side is a Pierpont prime.\n, the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023.\n\n# Polygon construction\nIn the [mathematics of paper folding](https://bliptext.com/articles/mathematics-of-paper-folding), the [Huzita axioms](https://bliptext.com/articles/huzita-hatori-axioms) define six of the seven types of fold possible.\nIt has been shown that these folds are sufficient to allow the construction of the points that solve any [cubic equation](https://bliptext.com/articles/cubic-equation).\nIt follows that they allow any [regular polygon](https://bliptext.com/articles/regular-polygon) of N sides to be formed, as long as\nN ≥ 3\nand is of the form\n2m3nρ\n, where ρ is a product of distinct Pierpont primes.\nThis is the same class of regular polygons as those that can be constructed with a [compass](https://bliptext.com/articles/compass), [straightedge](https://bliptext.com/articles/straightedge), and [angle trisector](https://bliptext.com/articles/angle-trisection).\nRegular polygons which can be constructed with only compass and straightedge ([constructible polygons](https://bliptext.com/articles/constructible-polygon)) are the special case where\nn = 0\nand ρ is a product of distinct [Fermat primes](https://bliptext.com/articles/fermat-prime), themselves a subset of Pierpont primes.\nIn 1895, [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician) studied the same class of regular polygons; his work is what gives the name to the Pierpont primes.\nPierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw [conic sections](https://bliptext.com/articles/conic-section) whose coefficients come from previously constructed points.\nAs he showed, the regular N-gons that can be constructed with these operations are the ones such that the [totient](https://bliptext.com/articles/euler-s-totient-function) of N is 3-smooth.\nSince the totient of a prime is formed by subtracting one from it, the primes N for which Pierpont's construction works are exactly the Pierpont primes.\nHowever, Pierpont did not describe the form of the composite numbers with 3-smooth totients.\nAs Gleason later showed, these numbers are exactly the ones of the form\n2m3nρ\ngiven above.\nThe smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the [hendecagon](https://bliptext.com/articles/hendecagon) is the first regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections).\nAll other regular N-gons with\n3 ≤ N ≤ 21\ncan be constructed with compass, straightedge and trisector.\n\n# Generalization\nA **Pierpont prime of the second kind** is a prime number of the form 2u3v − 1.\nThese numbers are\nThe largest known primes of this type are [Mersenne primes](https://bliptext.com/articles/mersenne-prime); currently the largest known is (41,024,320 decimal digits).\nThe largest known Pierpont prime of the second kind that is not a Mersenne prime is, found by [PrimeGrid](https://bliptext.com/articles/primegrid).\nA **generalized Pierpont prime** is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. A **generalized Pierpont prime** of the second kind is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. Since all primes greater than 2 are [odd](https://bliptext.com/articles/parity-mathematics), in both kinds p1 must be 2.\nThe sequences of such primes in the [OEIS](https://bliptext.com/articles/oeis) are:","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:15:09.444Z","updated_at":"2025-01-25T22:15:09.444Z","search_vector":"'pierpont':1 'prime':2","search_content":" \n**Named After:** [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician) \n**Terms Number:** Thousands \n**Con Number:** Infinite \n**Parentsequence:** Pierpont number \n**First Terms:** 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889 \n**Largest Known Term:** 81 × 220,498,148 + 1 \n**Oeis:** A005109 \n \n\n# \nIn [number theory](https://bliptext.com/articles/number-theory), a **Pierpont prime** is a [prime number](https://bliptext.com/articles/prime-number) of the form\nfor some nonnegative [integers](https://bliptext.com/articles/integer) u and v. That is, they are the prime numbers p for which\np − 1\nis [3-smooth](https://bliptext.com/articles/smooth-number).\nThey are named after the mathematician [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician), who used them to characterize the [regular polygons](https://bliptext.com/articles/regular-polygon) that can be constructed using [conic sections](https://bliptext.com/articles/conic-section).\nThe same characterization applies to polygons that can be constructed using ruler, compass, and [angle trisector](https://bliptext.com/articles/angle-trisection), or using [paper folding](https://bliptext.com/articles/mathematics-of-paper-folding).\nExcept for 2 and the [Fermat primes](https://bliptext.com/articles/fermat-prime), every Pierpont prime must be 1 modulo 6.\nThe first few Pierpont primes are:\nIt has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.\n\n# Distribution\nA Pierpont prime with\nv = 0\nis of the form 2^u+1, and is therefore a [Fermat prime](https://bliptext.com/articles/fermat-prime) (unless\nu = 0\n). If v is [positive](https://bliptext.com/articles/positive-number) then u must also be [positive](https://bliptext.com/articles/positive-number) (because 3^v+1 would be an [even number](https://bliptext.com/articles/even-number) greater than 2 and therefore not prime), and therefore the non-Fermat Pierpont primes all have the form\n6k + 1\n, when k is a positive integer (except for 2, when\nu = v = 0\n).\nEmpirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 106, 65 less than 109, 157 less than 1020, and 795 less than 10100.\nThere are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the [Mersenne prime](https://bliptext.com/articles/mersenne-prime) condition that the exponent must be prime.\nThus, it is expected that among n-digit numbers of the correct form, the fraction of these that are prime should be proportional to\n1/n\n, a similar proportion as the proportion of prime numbers among all n-digit numbers.\nAs there are numbers of the correct form in this range, there should be \\Theta(n) Pierpont primes.\n[Andrew M. Gleason](https://bliptext.com/articles/andrew-m-gleason) made this reasoning explicit, [conjecturing](https://bliptext.com/articles/conjecture) there are infinitely many Pierpont primes, and more specifically that there should be approximately\n9n\nPierpont primes up to\n10n\n. According to Gleason's conjecture there are Pierpont primes smaller than N, as opposed to the smaller conjectural number of Mersenne primes in that range.\n# [Distribution of the exponents for the smaller Pierpont primes | upload.wikimedia.org/wikipedia/commons/a/a0/Pierpont///exponent///distribution.png]\n\n# Primality testing\nWhen 2^u > 3^v, is a [Proth number](https://bliptext.com/articles/proth-number) and thus its primality can be tested by [Proth's theorem](https://bliptext.com/articles/proth-s-theorem).\nOn the other hand, when 2^u < 3^v alternative primality tests for are possible based on the factorization of M-1 as a small even number multiplied by a large [power of 3](https://bliptext.com/articles/power-of-3).\n\n# Pierpont primes found as factors of Fermat numbers\nAs part of the ongoing worldwide search for factors of [Fermat numbers](https://bliptext.com/articles/fermat-number), some Pierpont primes have been announced as factors.\nThe following table gives values of m, k, and n such that\nThe left-hand side is a Fermat number; the right-hand side is a Pierpont prime.\n, the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023.\n\n# Polygon construction\nIn the [mathematics of paper folding](https://bliptext.com/articles/mathematics-of-paper-folding), the [Huzita axioms](https://bliptext.com/articles/huzita-hatori-axioms) define six of the seven types of fold possible.\nIt has been shown that these folds are sufficient to allow the construction of the points that solve any [cubic equation](https://bliptext.com/articles/cubic-equation).\nIt follows that they allow any [regular polygon](https://bliptext.com/articles/regular-polygon) of N sides to be formed, as long as\nN ≥ 3\nand is of the form\n2m3nρ\n, where ρ is a product of distinct Pierpont primes.\nThis is the same class of regular polygons as those that can be constructed with a [compass](https://bliptext.com/articles/compass), [straightedge](https://bliptext.com/articles/straightedge), and [angle trisector](https://bliptext.com/articles/angle-trisection).\nRegular polygons which can be constructed with only compass and straightedge ([constructible polygons](https://bliptext.com/articles/constructible-polygon)) are the special case where\nn = 0\nand ρ is a product of distinct [Fermat primes](https://bliptext.com/articles/fermat-prime), themselves a subset of Pierpont primes.\nIn 1895, [James Pierpont](https://bliptext.com/articles/james-pierpont-mathematician) studied the same class of regular polygons; his work is what gives the name to the Pierpont primes.\nPierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw [conic sections](https://bliptext.com/articles/conic-section) whose coefficients come from previously constructed points.\nAs he showed, the regular N-gons that can be constructed with these operations are the ones such that the [totient](https://bliptext.com/articles/euler-s-totient-function) of N is 3-smooth.\nSince the totient of a prime is formed by subtracting one from it, the primes N for which Pierpont's construction works are exactly the Pierpont primes.\nHowever, Pierpont did not describe the form of the composite numbers with 3-smooth totients.\nAs Gleason later showed, these numbers are exactly the ones of the form\n2m3nρ\ngiven above.\nThe smallest prime that is not a Pierpont (or Fermat) prime is 11; therefore, the [hendecagon](https://bliptext.com/articles/hendecagon) is the first regular polygon that cannot be constructed with compass, straightedge and angle trisector (or origami, or conic sections).\nAll other regular N-gons with\n3 ≤ N ≤ 21\ncan be constructed with compass, straightedge and trisector.\n\n# Generalization\nA **Pierpont prime of the second kind** is a prime number of the form 2u3v − 1.\nThese numbers are\nThe largest known primes of this type are [Mersenne primes](https://bliptext.com/articles/mersenne-prime); currently the largest known is (41,024,320 decimal digits).\nThe largest known Pierpont prime of the second kind that is not a Mersenne prime is, found by [PrimeGrid](https://bliptext.com/articles/primegrid).\nA **generalized Pierpont prime** is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. A **generalized Pierpont prime** of the second kind is a prime of the form with k fixed primes p1 < p2 < p3 < ... < pk. Since all primes greater than 2 are [odd](https://bliptext.com/articles/parity-mathematics), in both kinds p1 must be 2.\nThe sequences of such primes in the [OEIS](https://bliptext.com/articles/oeis) are:","revisionCount":0},"hasMore":false,"nextCursor":null}

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