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A325143 Primes represented by cyclotomic binary forms. 4
3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 109, 113, 127, 137, 139, 149, 151, 157, 163, 173, 181, 193, 197, 199, 211, 223, 229, 233, 241, 257, 269, 271, 277, 281, 283, 293, 307, 313, 317, 331, 337, 349, 353, 367, 373 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A cyclotomic binary form over Z is a homogeneous polynomial in two variables which has the form f(x, y) = y^EulerPhi(k)*CyclotomicPolynomial(k, x/y) where k is some integer >= 3. An integer n is represented by f if f(x, y) = n has an integer solution.

LINKS

Peter Luschny, Table of n, a(n) for n = 1..10000

√Čtienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, Acta Arithmetica 184 (2018), 67-86; arXiv:1712.09019, arXiv:1712.09019 [math.NT], 2017.

PROG

(Julia) using Nemo

function isA325143(n)

    (n < 3 || !isprime(ZZ(n))) && return false

    R, x = PolynomialRing(ZZ, "x")

    K = floor(Int, 5.383*log(n)^1.161) # Bounds from

    M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt

    N = QQ(n)

    for k in 3:K

        e = Int(eulerphi(ZZ(k)))

        c = cyclotomic(k, x)

        for m in 1:M, j in 0:M if max(j, m) > 1

            N == m^e*subst(c, QQ(j, m)) && return true

    end end end

    return false

end

[n for n in 1:373 if isA325143(n)] |> println

CROSSREFS

Subsequence of A296095. Complement A325145. Number of A325141.

Cf. A293654, A299214, A299498, A299733, A299928, A299930, A299956, A299964.

Sequence in context: A123567 A059645 A090190 * A276357 A065041 A065393

Adjacent sequences:  A325140 A325141 A325142 * A325144 A325145 A325146

KEYWORD

nonn

AUTHOR

Peter Luschny, May 16 2019

STATUS

approved

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Last modified June 20 02:51 EDT 2021. Contains 345157 sequences. (Running on oeis4.)