A299930 -id:A299930 - cp's OEIS frontend

A299956 Prime numbers not in A299930.

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 113, 131, 137, 149, 151, 157, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 331, 337, 347, 353, 359

Offset: 1

Peter Luschny, Feb 25 2018

nonn

These prime numbers cannot be represented by a cyclotomic binary form f as p = f(x,y) with x and y odd prime numbers and x > y. Apart from 2 they appear as the leafs of cbf-trees introduced in A299930.

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299964.

isA299956(n) = isprime(ZZ(n)) && !isA299930(n)
A299956list(upto) = [n for n in 1:upto if isA299956(n)]
println(A299956list(1000))

A299733 Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.

Original entry on oeis.org

19, 97, 33751

Offset: 1

Peter Luschny, Feb 21 2018

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.

There are only three prime numbers below 600000 which satisfy the given conditions. No prime number below 600000 exists which has more than one representation if we require a representation by odd prime numbers y < x.

			33751 = f(131,79) for f(x,y) = x^2 + x*y + y^2.
33751 = f( 13, 2) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

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

Subsequence of A299929.

Cf. A293654, A296095, A299214, A299498, A299928, A299930, A299956, A299964.

using Nemo
function isA299733(n)
    if n < 3 || !isprime(ZZ(n)) return false end
    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); multi = 0
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 2:M if isprime(ZZ(m))
            for j in m:M if isprime(ZZ(j))
                if N == m^e*subst(c, QQ(j,m)) multi += 1
    end end end end end end
    multi > 1
end # Peter Luschny, May 16 2019

A299733(upto) =
{
    my(K, M, phi, multi);
    forprime(n = 2, upto, multi = 0;
        K = floor(5.383*log(n)^1.161);
        M = floor(2*sqrt(n/3));
        for(k = 3, K,
            phi = eulerphi(k);
            forprime(y = 2, M,
                forprime(x = y + 1, M,
                    if(n == y^phi*polcyclo(k, x/y),
                        multi += 1
                    )
                )
            )
        );
        if(multi > 1, print(n," has multiple reps!"))
    )
}
A299733(100000)

A299498 Integers primitively represented by cyclotomic binary forms.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 17, 19, 21, 25, 26, 29, 31, 34, 37, 39, 41, 43, 49, 50, 53, 55, 57, 58, 61, 65, 67, 73, 74, 79, 82, 85, 89, 91, 93, 97, 101, 103, 106, 109, 111, 113, 121, 122, 125, 127, 129, 130, 133, 137, 139, 145, 146, 147, 149, 151, 157, 163, 169, 170

Offset: 1

Peter Luschny, Feb 25 2018

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 primitively represented by f if f(x,y) = n has an integer solution such that x is prime to y.

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

Cf. A293654, A296095, A299214, A299733, A299928, A299929, A299930, A299956, A299964.

using Nemo
function isA299498(n)
    isPrimeTo(n, k) = gcd(ZZ(n), ZZ(k)) == ZZ(1)
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(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 m+1:M if isPrimeTo(m, j)
            N == m^e*subst(c, QQ(j,m)) && return true
    end end end
    return false
end
A299498list(upto) = [n for n in 1:upto if isA299498(n)]
print(A299498list(170))

A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.

Original entry on oeis.org

7, 13, 19, 29, 34, 37, 39, 49, 53, 55, 58, 61, 67, 74, 79, 91, 93, 97, 103, 109, 125, 127, 129, 130, 139, 146, 147, 163, 170, 173, 178, 194, 199, 201, 211, 217, 218, 219, 223, 229, 237, 247, 259, 273, 277, 283, 290, 291, 293, 298, 309, 313, 314, 327, 338, 349

Offset: 1

Peter Luschny, Feb 21 2018

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.

			There are exactly four ways to represent 13 by a cyclotomic binary form f(x,y) if we require x > y > 0. In one case, x and y are prime.
13 = f(2, 1) where f(x, y) = x^4 - x^2*y^2 + y^4,
13 = f(3, 1) where f(x, y) = x^2 + x*y + y^2,
13 = f(3, 2) where f(x, y) = x^2 + y^2,
13 = f(4, 3) where f(x, y) = x^2 - x*y + y^2.

Trygve Nagell, Sur les représentations de l’unité par les formes binaires biquadratiques du premier rang, Arkiv för Mat. 5 (6), (1965), 477-521, (p. 517).

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

Cf. A299929 (represented primes), A293654, A296095, A299214, A299498, A299733, A299930, A299956, A299964.

using Nemo
function isA299928(n)
    R, z = PolynomialRing(ZZ, "z")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
    N = QQ(n)
    P(u) = (p for p in u:M if isprime(ZZ(p)))
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, z)
        for y in P(2), x in P(y+1)
            N == y^e*subst(c, QQ(x, y)) && return true
        end
    end
    return false
end
A299928list(upto) = [n for n in 1:upto if isA299928(n)]
println(A299928list(350))

A299964 Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.

Original entry on oeis.org

19, 39, 97, 147, 247, 259, 327, 399, 410, 427, 481, 650, 777, 890, 903, 1010, 1027, 1130, 1209, 1267, 1443, 1490, 1533, 1677, 1730, 1767, 1802, 1813, 1898, 1911, 1970, 2037, 2119, 2210, 2330, 2378, 2667, 2793, 2847, 3050, 3170, 3297, 3367, 3477, 3530, 3603

Offset: 1

Peter Luschny, Feb 25 2018

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 in this sequence if f(x,y) = n has more than one integer solution where f is a cyclotomic binary form and x and y are prime numbers with 0 < y < x.

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

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299930, A299956.

function countA299928(n)
    R, z = PolynomialRing(ZZ, "z")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
    N = QQ(n); count = 0
    P(u) = (p for p in u:M if isprime(ZZ(p)))
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, z)
        for y in P(2), x in P(y+1)
            if N == y^e*subst(c, QQ(x, y))
                count += 1
    end end end
    return count
end
A299964list(upto) = [n for n in 1:upto if countA299928(n) > 1]
println(A299964list(3640))

A299929 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x.

Original entry on oeis.org

7, 13, 19, 29, 37, 53, 61, 67, 79, 97, 103, 109, 127, 139, 163, 173, 199, 211, 223, 229, 277, 283, 293, 313, 349, 397, 421, 433, 439, 457, 463, 487, 541, 577, 607, 641, 643, 691, 727, 733, 739, 787, 877, 937, 997, 1009, 1031, 1063, 1093, 1327, 1373, 1423, 1447

Offset: 1

Peter Luschny, Feb 21 2018

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.

			6841 = f(7,5) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

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

Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299930, A299956, A299964.

A299929list(upto) = [n for n in 1:upto if isprime(ZZ(n)) && isA299928(n)]
println(A299929list(1450))

isA299929[n_] := If[! PrimeQ[n], Return[False],
   K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
   For[k = 3, k <= K, k++,
   For[y = 1, y <= M, y++, If[PrimeQ[y], For[x = y + 1, x <= M, x++, If[PrimeQ[x],
   If[n == y^EulerPhi[k] Cyclotomic[k, x/y], Return[True]]]]]]];
Return[False]]; Select[Range[1450], isA299929]

A325143 Primes represented by cyclotomic binary forms.

Original entry on oeis.org

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

A325145 Primes not representable by cyclotomic binary forms.

Original entry on oeis.org

2, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Peter Luschny, May 16 2019

nonn

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.

A325143 gives the primes represented by cyclotomic binary forms.

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

[n for n in 1:1223 if isprime(ZZ(n)) && ! isA325143(n)] |> println

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))

cp's OEIS Frontend

A299956 Prime numbers not in A299930.

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A299733 Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.

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PARI

A299498 Integers primitively represented by cyclotomic binary forms.

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A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.

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A299964 Integers represented in more than one way by a cyclotomic binary form f(x,y) where x and y are prime numbers and 0 < y < x.

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A299929 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x.

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Mathematica

A325143 Primes represented by cyclotomic binary forms.

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A325145 Primes not representable by 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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