A325145 -id:A325145 - cp's OEIS frontend

A325143 Primes represented by cyclotomic binary forms.

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

Offset: 1

Peter Luschny, May 16 2019

nonn

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.

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

Subsequence of A296095. Complement A325145. Number of A325141.

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

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

A325870 Primes represented by non-quadratic cyclotomic binary forms.

Original entry on oeis.org

11, 13, 17, 31, 43, 61, 73, 97, 127, 151, 181, 193, 211, 241, 257, 331, 337, 421, 461, 463, 521, 541, 547, 577, 601, 641, 683, 757, 881, 991, 1009, 1021, 1031, 1093, 1297, 1621, 1801, 1871, 1873, 1933, 2221, 2417, 2657, 2731, 2801, 3001, 3121, 3361, 3571, 3697

Offset: 1

Peter Luschny, May 26 2019

nonn

Peter Luschny, Table of n, a(n) for n = 1..185
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.

Cf. A296095, A293654, A325143, A325145.

isA325870(n) =
{
    my(K, M, phi);
    K = floor(5.383*log(n)^1.161);
    M = floor(2*sqrt(n/3));
    for(k = 3, K,
        phi = eulerphi(k);
        if(phi >= 4,
            for(y = 1, M,
                for(x = y + 1, M,
                    if(n == y^phi*polcyclo(k, x/y),
                        return(1)
    )))));
    return(0)
}

At the suggestion of Michel Waldschmidt

cp's OEIS Frontend

A325143 Primes represented by cyclotomic binary forms.

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