A299214 -id:A299214 - cp's OEIS frontend

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.

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)

A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.

Original entry on oeis.org

19, 37, 79, 97, 109, 127, 139, 163, 223, 229, 277, 283, 313, 349, 397, 421, 433, 439, 457, 607, 643, 691, 727, 733, 739, 877, 937, 997, 1063, 1093, 1327, 1423, 1459, 1489, 1567, 1579, 1597, 1627, 1657, 1699, 1753, 1777, 1801, 1987, 1999, 2017, 2089, 2113, 2203

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 represented by f if f(x,y) = n has an integer solution.
We say a prime number p decomposes into x and y if x and y are odd prime numbers and there exists a cyclotomic binary form f such that p = f(x,y). The transitive closure of this relation can be displayed as a binary tree, the cbf-tree of p. A cbf-tree is squarefree if all its leafs are distinct. Examples are:
.
      33751                   23833                   310567
       / \                     /  \                    /  \
    131   79                163    19               359    283
          / \               / \    / \                     / \
         7   3            11   3  5   3                  19   13
                                                        /  \
                                                       5    3
.
The leaves of these trees are in A299956. Related to the question whether the root of a cbf-tree can be reconstructed from its leafs is A299733.

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

Cf. A299956 (complement), A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299956, A299964.

using Nemo
function isA299930(n)
    !isprime(ZZ(n)) && return false
    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(3), x in P(y+2)
            N == y^e*subst(c, QQ(x, y)) && return true
    end end
    return false
end
A299930list(upto) = [n for n in 1:upto if isA299930(n)]
println(A299930list(2203))

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))

A299956 Prime numbers not in A299930.

Original entry on oeis.org

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))

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

cp's OEIS Frontend

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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A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.

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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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A299956 Prime numbers not in A299930.

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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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