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A299928
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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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8
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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(Julia)
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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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