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.

464, 21, 3313, 17

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

Table of n, a(n) for n=1..4.

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 <a to b possible solutions m, n, using parity of m */
scy1(a, b, fr) =
{local(m, n, r, c, c1, c2, d, g, po, be, t1, t2, t3, st, dd);
st = 1; dd = 1; be = b-a; t1 = gettime();
for(m=a, b, po = polcyclo(m, t); if(m % 2 == 0, st = 2; dd = 3; ,
st = 1; dd = 2; );
forstep(n=dd, m-1, st, g = gcd(n, m);
if(g == 1, c1 = cy1(n); r = 2*m*n+1;
if(c1 % r == 0, c2 = cy(n, m); d = gcd(c2, c1);
if(d == r, print([m, n, r]);
); ); ); );
if(m % fr == 0,
t2 = gettime(); t3 = t2+t1; print([m, t3, ((m-a)/be)*100.0]);
t1 = t3;
); ); }

Crossrefs

Sequence in context: A108834 A233310 A220716 * A111697 A116344 A260497

Adjacent sequences: A189827 A189828 A189829 * A189831 A189832 A189833

Keyword

nonn,more

Author

Luis H. Gallardo, Apr 28 2011

Status

approved