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 A189830 Pairs of numbers i,j ordered by increasing i, such that 2 <= j < i, gcd(i,j)=1 and gcd(Phi_j(i), Phi_i(j))=2*i*j+1, where Phi_k(t) is the k-th cyclotomic polynomial. 0
 464, 21, 3313, 17 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The second pair i=3313, j=17 is the only known counterexample to a conjecture of Stephens that gcd(Phi_p(q), Phi_q(p))=1 for every pair of prime numbers (p,q). This is related to a conjecture of Feit-Thompson. See the corresponding wiki page. Up to i=9400 there are no new terms of the sequence. LINKS MathOverflow, Variant of Stephens result N. M. Stephens, On the Feit-Thompson Conjecture, Math. Comp. 25 (1971), 625. Wikipedia, Feit-Thompson conjecture EXAMPLE i=a(1)=464 and j=a(2)=21 since i=464 is the smallest positive integer such that gcd(Phi_i(j), Phi_j(i)) = 2*i*j+1 for a positive integer j such that 2 <= j < i and gcd(i,j)=1. PROG (PARI) /* define \$cy(m, n) = Phi_m(n)\$ the \$m\$-th cyclotomic polynomial evaluated at \$t=n\$ */ cy(m, n) = {local(po); po = polcyclo(m, t); subst(po, t, n); } /* for fixed \$m\$ compute \$cy(m, n)\$ */ cy1(n) = {subst(po, t, n); } /* search from m

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Last modified May 17 12:55 EDT 2021. Contains 343971 sequences. (Running on oeis4.)