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A299930
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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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9
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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
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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.
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:
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33751 23833 310567
/ \ / \ / \
131 79 163 19 359 283
/ \ / \ / \ / \
7 3 11 3 5 3 19 13
/ \
5 3
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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.
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LINKS
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PROG
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(Julia)
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))
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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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