A086397 Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.

3, 7, 41, 63018038201, 19175002942688032928599

Offset: 1

Author

Cino Hilliard, Sep 06 2003

Keywords

• frac
• more
• nonn

Comments

Next term, if it exists, is bigger than 489 digits (the 1279th convergent to sqrt(2)). - Joshua Zucker, May 08 2006

Are the terms >= 7 the primes in A183064? Is this a subsequence of A088165? - R. J. Mathar, Aug 16 2019

Yes, the terms >= 7 are the primes in A183064 and are a subsequence of A088165. a(1)=3 is from the numerator of 3/2, but no other convergents > sqrt(2) can appear in this sequence because they all have even denominator. All terms >= 7 are lower principal convergents from A002315/A088165/A183064 - Martin Fuller, Apr 08 2023

Links

• Andrej Dujella, Mirela Jukić Bokun, and Ivan Soldo, A Pellian equation with primes and applications to D(-1)-quadruples, arXiv:1706.01959 [math.NT], 2017.

Crossrefs

Denominators are A118612.

Programs

Mathematica

For[n = 2, n < 1500, n++, a := Join[{1}, Table[2, {i, 2, n}]]; If[PrimeQ[Denominator[FromContinuedFraction[a]]], If[PrimeQ[Numerator[FromContinuedFraction[a]]], Print[Numerator[FromContinuedFraction[a]]]]]] (* Stefan Steinerberger, May 09 2006 *)

PARI

cfracnumdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer)&&ispseudoprime(denom), print1(numer",");numer2=numer;denom2=denom); ) default(realprecision,28); }

Extensions

More terms from Cino Hilliard, Jan 15 2005

Edited by N. J. A. Sloane, Aug 06 2009 at the suggestion of R. J. Mathar