A378164 Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)*b*c)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.

1, 81, 1296, 2187, 1419857

Offset: 1

Author

Hugo Pfoertner, Nov 18 2024

Keywords

• nonn
• hard
• more

Comments

If it exists, a(6)>5*10^27.

Examples

			           Pairs b,c of consecutive
           terms of A076467
  A378165
    c-b        b        c     Quality = log(rad(c))/log(rad((c-b)*b*c))
      7,       1,       8,    0.262649535...
     44,      81,     125,    0.277532712...
     35,    1296,    1331,    0.309605475...
     10,    2187,    2197,    0.429917243...
  23040, 1419857, 1442897,    0.431260235...
		

Links

• Wikipedia, abc conjecture

Crossrefs

A378165 gives the corresponding values of c-b.

Cf. A007947 (rad), A076467, A377933, A377934, A378166, A378167.

Programs

PARI

\\ Uses M. F. Hasler's A076467_vec from A076467
a378164_5(upto) = {my(W=A076467_vec(upto), qw=0); for(k=2, #W, my(d=W[k]-W[k-1]); if(gcd([d,W[k],W[k-1]])==1, my(C=factor(W[k])[,1], B=factor(W[k-1])[,1], A=factor(d)[,1], P=vecprod(setunion(setunion(Set(B),Set(C)),Set(A))), q=log(vecprod(C))/log(P)); if(q>qw, print([d,W[k-1],W[k],q]); qw=q)))};
a378164_5(10^16)