A296095 -id:A296095 - 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

A300331 Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.

Original entry on oeis.org

5, 8, 9, 10, 11, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 40, 41, 45, 50, 53, 55, 58, 64, 65, 68, 72, 74, 81, 82, 85, 89, 90, 98, 100, 101, 104, 106, 113, 116, 122, 125, 128, 130, 136, 137, 144, 145, 146, 149, 153, 160, 162, 164, 170, 173, 176, 178, 180, 185

Offset: 1

Peter Luschny, Mar 06 2018

nonn

A cyclotomic binary form is a homogeneous polynomial in two variables of the form p(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function. An integer m is represented by p if p(x,y) = m has an integer solution.

m is in this sequence if and only if m is in A296095 but not in A300332. This means m can be represented by a cyclotomic binary form but not as m = Sum_{j in 0:p-1} x^j*y^(p-j-1) with p prime.

			1037 is in this sequence because 1037 = f(26,19) = f(29,14) with f(x,y) = y^2 + x^2 are the only representations of 1037 by a cyclotomic binary form (which has index 4).
1031 is not in this sequence because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4 (which has index 5).

Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A296095, A293654, A299930, A300332.

using Nemo
function isA300331(n)
    R, z = PolynomialRing(ZZ, "z")
    N = QQ(n)
    # Bounds from Fouvry & Levesque & Waldschmidt
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    r = false
    k = 2
    while k <= K
        if k == 7
            K = Int(ceil(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
            e = Int(eulerphi(ZZ(k)))
            c = cyclotomic(k, z)
            for y in 2:M, x in 1:y
                if N == y^e*subst(c, QQ(x, y))
                    isprime(ZZ(k)) && return false
                    r = true
                end
            end
        k += 1
    end
    return r
end
A300331list(upto) = [n for n in 1:upto if isA300331(n)]
println(A300331list(185))

A345894 Positive integers representable by the two cyclotomic binary forms Phi_5(x,y) and Phi_12(u,v).

Original entry on oeis.org

1, 16, 61, 81, 256, 625, 976, 1296, 2401, 4096, 4941, 6561, 10000, 14641, 15616, 20736, 28561, 38125, 38416, 50625, 65536, 79056, 83521, 104976, 130321, 146461, 160000, 194041, 194481, 229981, 234256, 249856, 279841, 331776, 390625, 400221, 456976, 531441

Offset: 1

Shashi Kant Pandey, Jul 23 2021

nonn

Positive integers C such that Phi_5(x,y) = Phi_12(u,v) = C has a solution with nonzero (x,y,u,v).

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.

			Phi_5(1,-3) = 1^4 + 1^3*(-3) + 1^2*(-3)^2 + 1*(-3)^3 + (-3)^4 = 1 - 3 + 9 - 27 + 81 = 61 and Phi_12(2, 3) = 2^4 - 2^2*3^2 + 3^4 = 16 - 36 + 81 = 61, so 61 is a term.

Étienne Fouvry, Claude Levesque and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Cf. A296095.

a(8)-a(38) from Jon E. Schoenfield, Jul 24 2021

cp's OEIS Frontend

A325143 Primes represented by cyclotomic binary forms.

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Julia

A325870 Primes represented by non-quadratic cyclotomic binary forms.

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A300331 Integers represented by a cyclotomic binary form Phi{k}(x,y) with positive integers x and y where max(x, y) >= 2 and the index k is not prime.

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Julia

A345894 Positive integers representable by the two cyclotomic binary forms Phi_5(x,y) and Phi_12(u,v).

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Extensions