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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. 9
 19, 97, 33751 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Table of n, a(n) for n=1..3. Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017. EXAMPLE 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. PROG (PARI) 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) (Julia) 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 CROSSREFS Subsequence of A299929. Cf. A293654, A296095, A299214, A299498, A299928, A299930, A299956, A299964. Sequence in context: A362301 A142170 A069593 * A086120 A129701 A221746 Adjacent sequences: A299730 A299731 A299732 * A299734 A299735 A299736 KEYWORD nonn,bref,more,hard AUTHOR Peter Luschny, Feb 21 2018 STATUS approved

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Last modified September 14 19:05 EDT 2024. Contains 375929 sequences. (Running on oeis4.)