A299214 Number of representations of integers by cyclotomic binary forms.

0, 0, 8, 16, 8, 0, 24, 4, 16, 8, 8, 12, 40, 0, 0, 40, 16, 4, 24, 8, 24, 0, 0, 0, 24, 8, 12, 24, 8, 0, 32, 8, 0, 8, 0, 16, 32, 0, 24, 8, 8, 0, 32, 0, 8, 0, 0, 12, 40, 12, 0, 32, 8, 0, 8, 0, 32, 8, 0, 0, 48, 0, 24, 40, 16, 0, 24, 8, 0, 0, 0, 4, 48, 8, 12, 24

Offset: 1

Author

Michel Waldschmidt, Feb 16 2018

Keywords

• nonn

Comments

a(m) is the number of solutions of the equation Phi_n(x,y) = m with n >= 3 and max{|x|,|y|} >= 2. Here the binary form Phi_n(x,y) is the homogeneous version of the cyclotomic polynomial phi_n(t).

One can prove that a(m) is always a multiple of 4.

Links

• Michel Waldschmidt, Table of n, a(n) for n = 1..1000
• Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

The sequence of indices m with a(m) != 0 is A296095.

The sequence of indices m with a(m) = 0 is A293654.

Programs

Julia

using Nemo
function countA296095(n)
    if n < 3 return 0 end
    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); count = 0
    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)) && (count += 1)
    end end end
    4*count
end
A299214list(upto) = [countA296095(n) for n in 1:upto]
print(A299214list(76)) # Peter Luschny, Feb 25 2018

Maple

x := 'x'; y := 'y':
with(numtheory): for n from 3 to 1000 do
F[n] := expand(y^phi(n)*cyclotomic(n, x/y))  od:
g := 0:
for m from 1 to 1000 do
   for n from 3 to 60 do  # For the bounds see the reference.
      for x from -60 to 60 do
         for y from -60 to 60 do
            if F[n] = m and  max(abs(x), abs(y)) > 1
                then g := g+1 fi:
         od:
      od:
   od: a[m] := g: print(m, a[m]): g := 0
od:

Mathematica

For[n = 3, n <= 100, n++, F[n] = Expand[y^EulerPhi[n] Cyclotomic[n, x/y]]]; g = 0; For[m = 1, m <= 100, m++, For[n = 3, n <= 60, n++, For[x = -60, x <= 60, x++, For[y = -60, y <= 60, y++, If[F[n] == m && Max[Abs[x], Abs[y] ] > 1, g = g+1]]]]; a[m] = g; Print[m, " ", a[m]]; g = 0];
Array[a, 100] (* Jean-François Alcover, Dec 01 2018, from Maple *)