A240469 Values k where the maximum number of distinct rational solutions to x^2 - Dy^2 = t, 0 < D <= k, 0 < t <= k, achieves a new record.

1, 2, 7, 10, 17, 32, 73, 144, 241, 336, 360, 720, 1080, 1260

Offset

1,2

Comments

Record values are in A240470.

Links

Table of n, a(n) for n=1..14.

Example

All Diophantine equations x^2 - Dy^2 = t, 0 < D <= 16, 0 < t <= 16, D squarefree, have fewer than 4 distinct solutions; the first with 4 solutions is x^2 - 17y^2 = 16 with the solutions (x,y) = (9/2,1/2), (21,5), (4,0), (13,3), so 17 is in the sequence.

Prog

(PARI) { r(l, k)=if(!issquarefree(l)||!polisirreducible(z^2-l), return(0)); v=bnfisintnorm(bnfinit(z^2-l), k); if(!#v, return(0)); s=0; for(k=1, #v, p=v[k]; a=polcoeff(p, 0); b=polcoeff(p, 1); f=1; for(l=k+1, #v, p=v[l]; aa=polcoeff(p, 0); bb=polcoeff(p, 1); if(abs(a)==abs(aa)&&abs(b)==abs(bb), f=0; break)); s=s+f); s
m=0; n=0; while(1, n=n+1; res=0; for(l=1, n, rr=r(l, n); if(rr>res, res=rr)); for(k=1, n-1, rr=r(n, k); if(rr>res, res=rr)); if(res>m, m=res; print(n, ", "))) }

Crossrefs

Cf. A218459, A077428, A035251, A031363.

Sequence in context: A140115 A294865 A105770 * A257335 A152211 A309805

Adjacent sequences: A240466 A240467 A240468 * A240470 A240471 A240472

Keyword

nonn,more

Author

Ralf Stephan, Apr 06 2014

Status

approved